8.6.4 Aerodynamic Forces on Compressor Blades

The conservation principles that we learned and applied to wings and bodies in external aerodynamics apply to internal aerodynamics as well. There are some complicating factors in a turbomachinery flow that are absent in external aerodynamics. The most dominant feature is the presence of strong swirl in turbomachinery that is either completely absent or weak in external flows. Second, the net turning in the flow is large in turbomachinery and zero in external aerodynamics. We may characterize the flow in a turbomachinery as lacking a unique flow direction, unlike external aerodynamics. As an example, the Kutta–Joukowski theorem on lift predicts a magnitude for an ideal two-dimensional lift (on a body that creates circulation, Γ) as ρ_∞V_∞Γ and a direction for the lift that is normal to V_∞. In turbomachinery blading, there are two distinct flow speeds, one upstream and the other downstream of the blade, that is, W₁ and W₂ and not just a single V_∞. Also, on the question of the direction of this force, is it normal to W₁ or W₂? In turbomachinery, we will define a “mean” flow angle and a “mean” velocity that help us describe the blade lift and its direction. There are other complicating factors in internal flows that deal with flow distortion (e.g., upstream wake transport and interaction with downstream blades) and unsteadiness (i.e., from neighboring blade rows in relative rotation) that are inherent in a turbomachinery stage and have either no or weak counterparts in external aerodynamics. In a compressor, the flow is continually subjected to an adverse pressure gradient, as the static pressure rises along the flow direction. On the contrary, in external aerodynamics the flow is only locally subjected to adverse pressure gradient. Therefore, we may distinguish the two adverse pressure gradients experienced by the internal and external flows as one having a “global” and the second a “local” character, respectively. Another subtle, yet important, difference between external aerodynamics and internal flows with swirl (as in turbomachinery) deals with the stability of such flows to external disturbances. The distinguishing feature of having a mean swirl profile in a compressor leads to centrifugal instability waves that may grow and cause compressor stall. The closest external aerodynamic experience to a centrifugal instability comes from the vortex breakdown on a delta wing at high angle of attack. Kerrebrock (1977) has illuminated the behavior of instabilities in a swirling flow (in an annulus) that provides for a new understanding of disturbances in swirling flows in turbomachinery.

Figure 8.17 is reproduced here to help with the control surface definition and the application of conservation principles to the flow in a compressor blade row.

Continuity demands:

(8.84)

Making an assumption of negligible radial shift in the stream surface from entrance to exit of the blade, we may conclude that r₂ ≈ r₁ and therefore s₂ ≈ s₁, hence,

(8.85)

The momentum equation in the axial (or z-) direction demands

(8.86)

Assuming a constant axial throughflow speed, that is, W_z1 = W_z2, we get

(8.87)

Therefore, we conclude that the axial force acting on the blade is in negative z-direction, that is, it points in the upstream direction. The conservation of tangential momentum requires

(8.88)

Therefore, we conclude that the tangential force on the blade is also in the negative θ-direction. These two blade forces are shown in Figure 8.31.

We may resolve the axial and tangential blade forces into a lift and drag components but first we have to define an average flow direction through the blade row. We define a mean flow direction β_m based on the average swirl in the blade row, namely,

(8.89)

where the average swirl, W_θm is defined as

(8.90)

The lift is the sum of the projections of the axial and tangential forces in a direction normal to the “average” flow direction defined in Figure 8.31, that is,

(8.91)

Also the drag is the sum of the projections of the two forces in the direction of “average” flow, namely,

(8.92)

The lift and drag forces acting on a compressor blade are shown in Figure 8.32.

Therefore the lift force (per unit span) is

(8.93)

Assuming an incompressible fluid and applying the Bernoulli equation to the static pressure rise term in Equation 8.93, we get

(8.94a)

This equation simplifies to

(8.94b)

(8.94c)

Finally, the blade sectional lift force is expressed as

(8.94d)

From Equation 8.58, we may replace the product of blade spacing and the change in swirl velocity by the blade circulation Γ, i.e. Γ ≡ s(W_θ1 − W_θ2), to get the familiar Kutta–Joukowski theorem counterpart in turbo-machinery flow, namely,

(8.95)

The first term is the familiar Kutta–Joukowski lift, and the second term is the effect of boundary layer formation and viscous losses on lift.

From Equation 8.92, we may write the blade drag force per unit span as

(8.96a)

Upon cancellation of the first two terms, we get a simple expression, entirely based on profile losses, for the blade drag force as

(8.96b)

The lift and drag coefficients are

(8.97)

where the cascade total pressure loss is nondimensionalized based on the inlet relative dynamic pressure and the σ is the blade solidity.

(8.98)

As expected, the blade two-dimensional drag coefficient in the limit of low-speed flow is related to the total pressure loss in the cascade, that is, the momentum deficit thickness in the blade wake. Furthermore, the drag coefficient is linearly proportional to the cascade total pressure loss measured in the wake, as expected.

We note that the wake-related loss in lift, in Equation 8.97, is proportional to the blade drag coefficient, Equation 8.98, therefore, the relation between the two-dimensional lift and drag coefficients may be written as

(8.99)

The first term in Equation 8.98 is the ideal lift coefficient, following the thin airfoil theory, and the second term accounts for the loss of lift due to boundary layer/wake formation. These expressions for the lift and drag coefficients, however, lack compressibility effects, three-dimensional or spanwise effects, and the end wall effects as in a real compressor. In addition, we remember that the actual flow is unsteady and involves wake transport and vortex shedding, which are not modeled here. However, we may take additional steps in correcting for some of the cited shortcomings. From Prandtl’s classical wing theory, we remember that the drag due to lift, or expressed as the induced drag coefficient C_Di, was related to the wing lift coefficient following

(8.100)

where the span efficiency factor e in the denominator represents the nonelliptic lift contribution/penalty and the term AR represents the wing aspect ratio. A distinguishing feature of the turbomachinery wake is its spiral shape rather than the flat wake of the Prandtl’s lifting line theory, where Equation 8.100 is derived. Second, the presence of solid walls acts as mirrors for the vortices (i.e., they create images), and the total induced flow has to account for the image vortices as well. A simple diagram of a vortex next to a wall and its image that renders the wall a stream surface (i.e., flow tangency condition on the wall is satisfied) is shown in Figure 8.33. The vortex strength is shown as Γ and the image vortex strength is, thus (–Γ) and is equally disposed on the opposite side of the wall. If we calculate the induced velocity at the wall due to these two vortices, the normal component to the wall vanishes, due to symmetry and the tangential component to the wall is doubled, again due to symmetry (Figure 8.33b).

The presence of the image vortex with a counter swirl is to reduce the effective induced velocity at the blades, as compared with a vortex filament in an unbounded space. Therefore, the simple expression for the drag polar for a wing, which is the sum of the profile drag and the induced drag

(8.101)

should be replaced by a more complex structure in a turbomachinery blading such as

(8.102)

In Equation 8.102, κ represents the influence factor on the induced drag due to the presence of solid walls (i.e., the annulus) and helical wakes, the end wall drag coefficient is due to viscous losses and the vortex formation (tip clearance and scraping vortex) near the wall, the drag due to shock is accounted for through C_{D, shock} Turbulent mixing and the unsteady drag due to vortex shedding (the root-mean square) are reflected in the last term of Equation 8.102. Despite its rather complete look, Equation 8.102 is just an empirical model, which requires an extensive experimental database in order to be useful as a predictive tool. However, breaking down the blade loss to its basic constituent levels helps our understanding of the complex flow phenomena in turbo-machinery. We will address three-dimensional losses, compressibility effects, and unsteadiness further in this chapter.

Our discussion of compressor aerodynamics has been limited to 2D flows. The behavior trend of thermodynamic and flow variables in a compressor stage with an IGV in two dimensions are reviewed in Figure 8.34. For additional reading on cascade aerodynamics, Gostelow (1984) should be consulted.

8.6.5 Three-Dimensional Flow

Our discussion of the flowfield so far has been limited to two-dimensional flows represented by the “pitchline” radius. The next level of complexity brings us to the three-dimensional flow analysis with a host of simplifying assumptions. These assumptions are necessary to render the analysis tenable. To relieve any of the assumptions requires numerical integration of the governing equations, which are coupled and nonlinear. The method of approach, in that case, is the computational fluid dynamics applied to a single or multiple stages in a compressor. Simplifying assumptions for our three-dimensional flow analysis are

1. Flow is steady, that is, ∂/∂t → 0
2. Flow is axisymmetric, that is, ∂/∂θ → 0
3. Flow is adiabatic, that is, q_wall = 0
4. Fluid is inviscid and nonheat conducting, that is, μ = 0 and κ = 0, respectively
5. Blade geometry has zero tip clearance, that is, blades stretch between hub and casing

In addition, we may choose to limit the radial shift of the stream surfaces to within the blade rows and hence require radial equilibrium both upstream and downstream of the blade rows. By limiting the radial shift of the stream surfaces to within the blade rows, we are thus creating cylindrical stream surfaces outside the blade rows, that is, C_r = 0 both upstream and downstream of the blade row. By assuming an axisymmetric flow, we are smearing out the blade-to-blade flow and thus lose blade periodicity for a finite number of blades. This is akin to having infinitely many blades. For the steady flow, we are placing the coordinate system with the blade row that operates in isolation in a cylindrical duct. The implication is that having a neighboring blade row in relative rotation with respect to the blade row of interest would render the flowfield unsteady. In the limit of adiabatic flow, the only mechanism for energy transfer to/from the fluid is the mechanical shaft power. The assumption of inviscid and nonheat conducting fluid eliminates viscous and thermal boundary layer formations. In essence, the only mechanism for energy exchange and momentum transfer is the fluid pressure and not the fluid molecular viscosity and thermal conductivity. The combination of adiabatic and inviscid flow creates a fictitious isentropic environment for the flow interaction with the blades. Figure 8.35 shows cylindrical stream surfaces entering and leaving an isolated blade row in a cylindrical duct.

All parameters in this problem become a function of r only. Note that by taking a cylindrical duct downstream of the blade row, we are in essence decoupling the flow in the vicinity of the blade row from axial pressure gradients. As noted earlier, the axisymmetric flow takes out the θ-dependency; therefore the only independent parameter left is the radial position r. The velocity field in station 2, that is, downstream of the rotor, has zero radial velocity and a general swirl distribution C_θ(r), as well as an axial velocity distribution C_z(r) that needs to be determined from the conservation laws. It is customary to specify a desired swirl distribution downstream of a blade row and calculate the corresponding axial velocity distribution that is supported by the swirl profile. This method of approach is called the vortex design of turbomachinery blades. The pressure distribution p(r) is established by imposing a radial equilibrium condition on the stream surfaces, which simplifies to a balance of pressure gradient and the centrifugal force on a swirling fluid stream, that is,

(8.103)

From a known swirl distribution C_θ(r), we get the spanwise blade work distribution following the Euler turbine equation, namely,

(8.104)

However, the only unknown in the Equation 8.104 is the downstream radius r₂ corresponding to an upstream stream surface radius r₁. This is the crux of the problem of radial shift within the blade row, which needs to be determined. To establish the radial shift of the stream surfaces, Δr, we need to satisfy the continuity equation between the stream surfaces, from hub to tip or in our case from ψ₁ to ψ₅ upstream and downstream of the blade row. For swirl profiles that produce a constant axial velocity downstream of a blade row, the problem becomes very simple. In other cases, we need to integrate a nonuniform axial velocity and density profile in the spanwise direction to calculate the mass flow rates between the stream surfaces and thus establish the radial shift Δr.

8.6.5.1 Blade Vortex Design

The swirl velocity downstream of a blade at a given radius, namely pitchline, may be defined via a two-dimensional approach, as outlined in the previous sections. Anchoring the swirl profile at the pitchline, we may describe a variety of swirl profiles in the spanwise direction that could be analyzed and assessed for their practicality and utility. Some common swirl profiles assume a combination of a potential vortex, a solid body rotation, or, in general, a power profile for the swirl distribution, C_θ(r). The stagnation enthalpy in the absolute frame of reference may be written as

(8.105)

The radial derivative of the above equation yields,

(8.106)

Also, from Gibbs equation we relate the enthalpy gradient to pressure and entropy gradients as

(8.107)

Now, let us substitute Equation 8.107 into Equation 8.106, to get an interim result, that is,

(8.108)

A constant-work blade will have the first term in Equation 8.108 vanish, otherwise the work distribution along the blade depends on the angular momentum distribution upstream and downstream of the blade according to Euler turbine equation. The entropy gradient term on the RHS of Equation 8.108 is the blade loss profile in the radial direction. This parameter has to be an input to the problem based on the accumulated loss data published in the literature or the proprietary data of the industry. It is common to ignore the radial loss profiles for the first attempt in establishing the density, velocity gradients downstream of a blade row. The static pressure gradient follows the radial equilibrium theory, namely, Equation 8.103. The axial velocity profile is the third term on the RHS of Equation 8.108, which is the goal of our analysis. The last term is the swirl profile that we consider an assumed function of radial position. Therefore, to establish an axial velocity profile C_z(r), we need to integrate

(8.109)

The last two terms in Equation 8.109 may be combined into a single derivative involving angular momentum to get

(8.110)

The special cases of swirl profile that are considered in compressor blade design are described in the next section.

Case 1: Free-vortex design

(8.111)

This is known as the potential vortex or free-vortex design. The name association “potential vortex” is due to the similarity between this swirl profile and that of a vortex filament (C_θ ∼1/r). We also remember that the flowfield about a vortex filament is irrotational, hence a potential vortex. The irrotational flow around a vortex filament has zero vorticity, hence the name free-vortex design.

Let us consider two scenarios here. First, consider the case of zero preswirl upstream of the rotor, that is, the case of no inlet guide vane, and the second scenario with an inlet guide vane that induces a solid-body rotation swirl profile of its own upstream of the rotor.

With no IGV, the flow upstream of the rotor is swirl free, that is,

(8.112)

The rotor inducing a free vortex swirl distribution in plane 2, gives

(8.113)

The Euler turbine equation demands

(8.114)

The swirl profile associated with a free-vortex design blade produces a constant work along the blade span. This is known as a constant-work rotor. Therefore, the blade work profile and the angular momentum profile terms in Equation 8.110 vanish, that is,

(8.115)

The axial velocity profile equation reduces to

(8.116)

Here, we have a choice and that is to either specify a radial loss profile from the available data in the literature or neglect the loss term in this preliminary phase of establishing an axial velocity profile. Here, we take the second route, although we shall present and discuss radial distribution of losses associated with modern fan blades later in the chapter.

In the limit of a loss-free rotor and the free-vortex design, we conclude that the axial velocity C_z remains constant, that is, Equation 8.116 yields,

(8.117)

Figure 8.36 shows the velocity pattern across the rotor.

In Figure 8.36, we have kept the casing radius constant and have increased the hub radius to account for the density rise across the blade row. There are other choices that we could take, for example, the casing could have been tapered and the hub remained cylindrical or both hub and casing could have been tapered. Since the calculation method is identical, we shall stay with our choice of constant tip radius and an increasing hub radius. It is also customary to design turbomachinery blades based on a constant throughflow speed, that is, axial velocity remains constant. Therefore, based on the continuity equation, the channel area develops inversely proportional to density ratio, assuming a uniform density profile upstream and downstream of the rotor, that is,

(8.118a)

Therefore, the ratio of r_h2 to r_t is related to the density ratio and the inlet hub-to-tip radius ratio,

(8.118b)

We tackle the density ratio problem after we examine other swirl profiles of interest. The swirl put in by the rotor needs to be taken out by the stator; therefore, we demand

(8.119)

which may be expressed in the shorthand form

(8.120)

where a is a constant and the plus sign describes the rotor swirl input to the flow and the minus sign designates the stator withdrawal of the swirl put in by the rotor. The exit flow in station 3 downstream of the stator is thus swirl free just as in the flow entering the stage.

The degree of reaction for this design is rewritten from Equation 8.46c, with a free vortex swirl distribution substituted for rotor exit swirl as

(8.121)

To cast this equation relative to the pitchline radius, we divide and multiply the second term in Equation 8.121 by to get

(8.122)

The term a/r_m is the rotor exit swirl velocity at the pitchline radius and the second term, that is, ωr_m, is the wheel speed at the pitchline radius. From Equation 8.122, we conclude that the product of these two terms is related to the degree of reaction at the pitchline via

(8.123)

Therefore, the degree of reaction for a compressor stage with no inlet guide vane and a free vortex swirl distribution produced by its rotor is

(8.124)

We have graphed this equation for three stage designs with values of pitchline degree of reaction specified at 0.5, 0.6, or 0.7. We immediately note that the rotor hub is in danger of very low degrees of reaction. Since a negative degree of reaction implies that the flow in that section is not compressing (i.e., that section behaves like a turbine!), we have limited our graphical presentation in Figure 8.37 to positive degree of reaction cases at the hub. For example, in the case of 50% degree of reaction at the pitchline the hub radius should be at ∼0.71 r_m, which calls for a minimum rotor with hub-to-tip radius ratio of ∼0.55. Any lower hub-to-tip radius ratio leads to a negative hub degree of reaction. For a typical value of a modern fan hub-tip radius ratio of ∼0.5, we need to raise the pitchline degree or reaction beyond 50%. A second source of concern is the rotor tip region, which is being asked to bear an unusually high share of the stage pressure rise.

The hub section of the rotor is to produce the highest swirl according to Figure 8.36. The tip section is imparting the lowest swirl. Thus, the free vortex swirl profile demands excessive turning of the flow near the hub and very small turning at the tip, which causes the free-vortex blades to be highly twisted. The attractiveness of the free-vortex design is in its simplicity of analysis, which leads to a constant-work rotor and constant axial velocity distribution downstream of blade rows.

Now, let us study the case of a compressor stage with an IGV that induces a preswirl of solid body rotation upstream of the rotor. The purpose of an inlet guide vane is to reduce the relative Mach number at the tip; hence, a free-vortex design that attains the lowest swirl velocity at the tip would make less sense than a solid body rotation type swirl distribution downstream of the IGV.

Therefore,

(8.125)

where b is a constant. Combining this swirl profile with a free-vortex distribution of the rotor, we get

(8.126)

The rotor work distribution is

(8.127a)

Substitute swirl profiles 8.125 and 8.126 in the rotor work input (Equation 8.127a) to get

(8.127b)

We conclude that the rotor work distribution along the rotor span is constant.

Now, let us substitute the swirl profile downstream of the rotor and the constant work distribution in Equation 8.110 to get an equation involving the axial velocity distribution,

(8.128)

This equation simplifies to

(8.129)

We may integrate this equation from a reference radius, for example, the pitchline radius r_m, where the axial velocity is known as C_zm, to any radius r to get

(8.130)

or in nondimensional form the axial velocity profile downstream of the rotor is

(8.131)

In order to calculate the downstream hub radius after the rotor, we need to integrate the product of density and axial velocity over the inlet and exit planes of the rotor, which are set equal via continuity equation. We need to ask whether the axial velocity downstream of the IGV, which is the rotor upstream, is uniform. We may use the same equation of radial equilibrium theory that we applied to the rotor and note that the IGV does no work on the fluid, hence the fluid total enthalpy remains constant. Consistent with our assumption of no radial loss in the preliminary stage of our blade row calculations, Equation 8.110 simplifies to

(8.132)

Upon integration, we get a parabolic axial velocity distribution upstream of the rotor, that is,

(8.133)

Let us compare the two axial velocity profiles upstream and downstream of the rotor by examining Equations 8.131 and 8.133. The axial velocity profile upstream of the rotor is parabolic, as noted and the profile downstream of the rotor has an extra logarithmic term, which tends to exacerbate the nonuniformity in the axial profile. This implies that the axial velocity at the tip is further reduced downstream of the rotor as compared with the upstream value. In contrast, the axial velocity near the hub shows an increase downstream of the rotor, hence an increased nonuniformity of the axial velocity profile. The swirl and axial velocity profiles across a compressor rotor that induces a free-vortex swirl distribution to an IGV flowfield with a solid-body rotation swirl profile are shown in Figure 8.38. The choice of solid-body rotation for the inlet guide vane is consistent with the desire to reduce the rotor tip Mach number; however, combining it with a free-vortex rotor leads to a highly nonuniform axial velocity profile, which is undesirable. What happens to the spanwise distribution of the degree of reaction in this case?

To answer this question, we recall that the swirl profiles across the rotor are

and

The equation for the degree of reaction is

(8.134)

We may cast this equation in terms of the pitchline radius, similar to Equation 8.124 as

(8.135)

In terms of the degree if reaction at the pitchline, °R_m, we may write Equation 8.135 as

(8.136)

To get the density profile downstream of an IGV, we differentiate the perfect gas law in the radial direction and substitute the centrifugal force for the radial pressure gradient to get

(8.138)

This equation may be solved for the density variation as

(8.139)

We may isolate dρ/ρ in terms of the known functions of r, namely

(8.140)

Now, we need to use the conservation of total enthalpy to write an expression for the static temperature distribution downstream of the IGV, that is,

(8.141)

The RHS of Equation 8.141 is a known function of the radial coordinate r via Equations 8.126 and 8.133. The differential of the static temperature may be written as

(8.142)

The logarithmic derivative of the static temperature is the combination of Equations 8.141 and 8.142 as

(8.143)

Substituting Equation 8.143 in 8.140, we may calculate the static density profile downstream of the IGV. This process is more accurate, but rather tedious, and is often simplified by making the following assumptions, namely,

(8.144)

(8.145)

Thus, to establish the hub radius downstream of the rotor, we satisfy the continuity equation via

(8.146)

The density profile downstream of the rotor is a function of temperature and velocity profile using the total enthalpy expression, similar to Equation 8.141. Again, the process is rather tedious and a simplification is in order. We know that the density and temperature are related via a polytropic exponent, namely,

(8.147)

where n is the polytropic exponent. For an isentropic flow, n = γ, and for irreversible adiabatic flows, which are encountered in turbomachinery n < γ. The density distribution is a function of the temperature distribution and the polytropic exponent, that is,

(8.148)

We may use the small stage or polytropic efficiency in a compressor, e_c, to relate pressure and density ratio to temperature ratio across a blade row as

(8.149)

(8.150)

Therefore the strategy is as follows. We first establish all parameters at the pitchline and then expand the results to other radii by using a blade vortex design choice, for example, free vortex. Therefore, we first establish the swirl at the pitchline using a criterion based on the diffusion factor or the degree of reaction at the pitchline. Then by using the Euler turbine equation, we get the absolute total temperature at the pitchline radius, T_t2m. The definition of total temperature at the pitchline and the velocity components leads to the static temperature downstream of the rotor at r_m, that is, T_2m. By using a value for compressor polytropic efficiency (say 0.90 or 0.91), we determine the static pressure and the density at r_m, that is, p_2m and ρ_2m. Now, all parameters are established at the pitchline radius. We are now ready to expand our results to other radii. We choose a blade vortex design type, which fits a swirl profile along the rotor span to the pitchline swirl, C_θm. Therefore, the blade vortex design choice establishes the radial distribution of swirl, that is, C_θ(r). With the knowledge of swirl downstream of the rotor, we repeat the above procedure at other radii to establish all thermodynamic and flow parameters. It is customary to repeat the procedure for an odd number of stream surfaces (5 or 7), which yield a pitchline stream surface as well. The minimum is thus three, that is, the hub, pitchline, and the tip stream surfaces. We calculate the axial velocity distribution downstream of the rotor, using Equation 8.110. With assumed compressor polytropic efficiency, say 0.91, we calculate the density ratio across the blade row using Equation 8.150. Finally, we conserve the mass flow rate on the two sides of the rotor to establish the annulus area downstream of the rotor. Repeat the same process for the stator. As the process of design is iterative by nature, we need to return to our initial choices of vortex design, D-factor at the pitchline, the degree of reaction at the pitchline, or simply the blade hub-to-tip radius ratio in order to achieve an acceptable preliminary design for the stage. For the choice of blades that produce the desired flow turning at a given radius without flow separation and with an adequate margin of safety, we need to rely on cascade data presented in Figure 8.22.

Case 2: Rotor with forced vortex design (no IGV)

The rotor induced swirl increases linearly with the radius, that is, of solid-body rotation type, hence,

(8.151)

The stator removes the swirl put in by the rotor, hence the stator exit plane is swirl free, that is, C_θ1 = C_θ3 = 0.

The work distribution along the rotor span is

(8.152)

Therefore, the rotor loading increases proportional to r², that is, the rotor tip is overworked and the hub is not sufficiently loaded. The overloading of the rotor tip could result in flow separation at the tip with excessive flow turning and high diffusion. Usually, the tip region in a modern fan operates in the supersonic range, which limits the amount of flow turning and the appearance of shocks at the blade trailing edge. We shall address the issues of supersonic blade sections in a transonic compressor in a later section of this chapter. The degree of reaction for this vortex profile is

(8.153)

The axial velocity profile may be derived from Equation 8.110 as

(8.154)

This equation is integrated with reference to the pitchline to produce

(8.155)

For ω = b, °R = 0.5 and the axial velocity remains uniform.

Figure 8.40 shows the spanwise distribution of the axial velocity downstream of the inlet guide vane with a solid-body rotation swirl profile. The spanwise maldistribution of axial velocity is of concern for the same reason as with the excessive tip flow turning, namely, large flow deceleration at the tip could exceed allowable diffusion limits (either through D-factor or C_p) and lead to stall.

Case 3: A general stage vortex design (with IGV)

Let us consider a blend of the free and forced vortex swirl distributions that may be generalized by

(8.156)

(8.157)

For n = 1, a combination of solid body rotation and a free-vortex swirl distribution are superimposed upstream and downstream of a compressor rotor. The function of stator, which is stated as to remove the swirl put in by the rotor, is seen from Equations 8.156 and 8.157.

The degree of reaction variation along the span may be written as

(8.158)

Obviously, the reaction distribution depends on the exponent n. For n = 1, the reaction along the stage height remains constant. This is desirable since the pressure rise along the blade becomes constant. For n > 1, the reaction decreases with span and for n < 1, the reaction increases with span. Any desired variation, including a constant, may be tailored through our choice of exponent, n of the swirl profile.

The rotor work distribution is

(8.159)

This result is independent of the exponent n in Equation 8.156. Therefore, we create a constant work rotor by the above swirl distribution, which is desirable. The combination of constant work, which means the total enthalpy rise across the rotor is uniform, and the constant reaction (for n = 1), which implies a constant static enthalpy rise across the rotor, leads to a constant kinetic energy downstream of the rotor. Since swirl profile is a function of r, then the axial velocity has to attain a profile with the spanwise direction. We made this argument to show that the axial velocity may not be uniform in this case.

The axial velocity profile follows the same approach as the previous cases, namely,

(8.160)

(8.161)

The case of n = 1 is of particular interest since it leads to a constant reaction. The axial velocity profile, for n = 1, is

(8.162)

The plus and minus in Equation 8.162 signify the upstream and downstream of the rotor, respectively. In general, the axial velocity decreases with radius with this choice of swirl profile.

8.6.5.2 Three-Dimensional Losses

The factors that render the flow process in a turbomachinery stage irreversible are

• End wall losses
  • Secondary flow losses
  • Tip clearance loss
  • Labyrinth seal and leakage flow losses
• Shock losses
  • Total pressure loss
  • Shock-boundary layer interaction
• Blade wake losses
  • Viscous profile drag (from cascade experiments)
  • Induced drag losses
  • Radial flow losses
• Unsteady flow losses
  • Upstream wake interaction
  • Vortex shedding in the wake
• Turbulent mixing.

Although the above list seems like a formidable assortment of flow losses, how many of them are totally decoupled from the rest? In a practical sense, the answer has to be none. Is there the possibility of duplication, that is, double bookkeeping, in the above list? The answer is, of course, there is a possibility of counting a loss (or a part of it) twice or three times. Therefore, we conclude that any broken down list of factors contributing to loss is artificial at best and needs to be viewed with caution. The “list” is made only as a tool to help our understanding of complex flow phenomena in broad categories, such as

• Compressibility effects
• Viscous and turbulent dissipation
• Unsteadiness
• Three dimensionality.

A transonic rotor creates a shock, which is weakened by suction surface expansion Mach waves and propagates upstream of the rotor row (Kerrebrock, 1981). Note that since the upstream flow is subsonic in the absolute frame of reference, there is no zone of silence preventing the propagation of the rotor created waves. Figure 8.41 shows a typical flowfield.

The broad category of “end wall” losses encompasses the annulus boundary layer, corner vortex, tip clearance flow, and the seal leakage flow. The tip clearance flow is a pressure-driven phenomenon that relieves the fluid on the pressure side towards the suction surface, as shown in Figure 8.42.

The boundary layer formation on the annulus is expected to grow with distance and in response to an adverse pressure gradient. The growth of the boundary layer with the number of stages is demonstrated by the data of Howell (1945), where the axial velocity distribution continually deforms and by the exit of the second stage there is no resemblance to a “potential core, ” that is, flat top profile, that the inlet flowfield characterizes. The formation of corner vortices and the associated total pressure loss may be discerned from Howell data, as shown in Figure 8.43. Note that the flow deflection angle near the hub and the tip of the cascade, in the boundary layer, is in excess of data outside the annulus boundary layer as seen in Figure 8.43b. Modern blade design accounts for this difference in the relative flow angle in the boundary layer and bends the blade ends (end-benders) to minimize incidence loss. A schematic drawing of an end-bender blade is shown in Figure 8.44. Howell (1945) proposes a simple model for end wall losses that relates an annulus drag coefficient C_Da to the ratio of blade spacing to blade height, that is,

(8.163)

A comparison paper by Howell (1945) on fluid dynamics of axial compressors should also be consulted. A secondary flow pattern is developed in a duct with a bend. The flow turning within the blade passages of a compressor then leads to a migration of the boundary layer fluid from the pressure surface toward the low-pressure side, that is, the suction surface. A flow pattern normal to the primary flow direction is then set up, which is called a secondary flow. A schematic drawing of a secondary flow generation and pattern is shown in Figure 8.45. Pioneering work on the formulation of secondary flows is due to Hawthorne (1951) and a simplified engineering formulation is due to Squire and Winter (1951). A simple secondary flow loss model is proposed by Howell (1945) and is expressed in terms of secondary (or induced) drag coefficient C_Ds,

(8.164)

The unsteadiness is an inherent mechanism for energy transfer between a rotating blade row and the fluid in turbomachinery. The upstream wake interaction with the following blade row is one source of unsteadiness (Kerrebrock and Mikolajczek, 1970). This wake chopping interaction leads to unsteady lift, which, following Kelvin’s circulation theorem, results in a shed vortex of opposite spin in the wake. Another source of unsteadiness to a blade row is its relative motion with respect to a downstream blade row. Although, the blade is not engaged in a viscous wake chopping activity, as in the previous case, it is operating in an unsteady pressure field created by the downstream blade row. Operating in the unsteady pressure field is then referred to as unsteady potential interaction. Blade vibration in bending, twist or combined bending, and torsion is inevitable for elastic cantilevered structures. Therefore, blade vibration induces a spanwise variation of the incidence angle, hence an unsteady lift with a subsequent vortex shedding in the wake. The unsteady vortex shedding phenomenon is schematically shown in Figure 8.46. Kotidis and Epstein (1991) have measured unsteady radial transport in the rotor wake of a transonic fan. They have related unsteady losses to the radial transport in the spanwise vortex cores as well as turbulent mixing. The individual losses and their contributions to the overall loss are depicted in Figure 8.47 (from Howell, 1945). We note that at the design point the overall losses are near minimum and in low flow, the compressor enters surge and at the high flow rates profile losses dominate due to flow separation.

8.6.5.3 Reynolds Number Effect

The flow environment in a compressor, due to adverse pressure gradient, is sensitive to the state of a boundary layer. A laminar boundary layer readily separates in an adverse pressure gradient environment whereas a turbulent boundary layer may sustain a large pressure rise without separation or stall. The blade chord length, c, represents a suitable length scale for the Reynolds number evaluation as well as the relative velocity to the blade, W. Due to nearly flat compressor blade surfaces, we may borrow concepts from the flat plate boundary layer theory to predict the behavior of the boundary layer in adverse pressure gradient on a compressor blade. The effects of unsteadiness inherent in a turbomachinery as well as higher levels of turbulence in the core flow tend to provide for a higher mixing at the boundary layer and thus enhance its ability to withstand pressure rise. A more accurate approach should also include the effect of the blade curvature and its destabilizing (on a concave surface) or stabilizing (on a convex surface, such as the suction surface of a blade) effect on the boundary layer development. A simple statement can be made regarding the state of the boundary layer and its relation to compressor loss and that is to postpone/avoid flow separation, the boundary layer on a compressor blade has to be turbulent. This poses a rough minimum Reynolds number based on the blade chord of ∼200, 000 that serves as a rule of thumb, that is,

The Reynolds number trend on the cascade exit flow angle and total pressure loss may be seen from cascade test data of Rhoden (1956) in Figure 8.48. We note that below Re_c ∼ 100, 000 the cascade suffers from a rapid increase in total pressure loss and large exit flow angle deviation, which is due to the phenomenon of laminar separation. This represents a lower critical Reynolds number. Koch (1981) examined the stalling pressure rise capability of axial-flow compressor stages, which indicated a much-reduced sensitivity to Reynolds number below 200, 000. The source of reduced sensitivity is found in higher turbulence levels and the unsteadiness inherent in a compressor stage. Koch’s data normalized to Re = 130, 000 is shown in Figure 8.49.

The boundary layers in transonic compressors readily reach the turbulent regime due to a high relative velocity W. For example, the Reynolds number on a fan blade operating at (relative) Mach 1.5 at the tip and 0.75 at the hub with a 1-ft (30.5 cm) chord at standard sea level conditions ranges from ∼5 × 10⁶ to ∼10⁷. For Re_c ∼ > 500, 000 the compressor efficiency remains nearly constant, which marks the upper critical Reynolds number. The blade surfaces behave as hydraulically rough and thus independent of Reynolds number for Re_c > 500, 000. At high altitudes, where the air density is low, and for a slow aircraft (e.g., a long-endurance observation platform), a suitable compressor blade should be designed with a much wider chord to combat the perils of laminar separation.

The effect of Reynolds number is expected to be minimal on secondary flow losses as the secondary flows are predominantly pressure-driven. Conventional, that is, steady, low turbulence, cascade tests do not reproduce compressor test rig results due to higher turbulence levels that promote mixing and the effect of unsteadiness with similar impact on the boundary layer. Finally, for the effect of Reynolds number on shock-boundary layer interaction, we may examine the work of Donaldson and Lange (1952). Figure 8.50 is a log–log graph of critical pressure rise across a shock that leads to boundary layer separation as a function of Reynolds number (from Donaldson and Lange). The two branches that appear on the graph correspond to the familiar laminar and turbulent boundary layers on a flat plate. We note that a turbulent boundary layer sustains nearly an order of magnitude (∼10 times) higher stalling pressure rise across the shock than a corresponding laminar boundary layer in a supersonic flow, as pointed out earlier in this section. Also note that the Reynolds number dependence for the stalling pressure rise in a turbulent flow ∼Re^–0.2 and for the laminar boundary layer is inversely proportional to the square root of Reynolds number, that is, ∼Re^–0.5, which is identical to the Reynolds number behavior of momentum deficit thickness or friction drag coefficient on flat plates.

8.7 Compressor Performance Map

Compressor pressure ratio plotted against the mass flow rate through the compressor is the compressor performance map. It is customary to graph the constant rpm lines on the performance chart as well as the adiabatic efficiency. The performance map of a single-stage transonic compressor is shown in Figure 8.51 (from Sulam, Keenan, and Flynn, 1970). The relative tip speed of the rotor is 1600 ft/s (488 m/s), and it represents a high-pressure ratio compressor stage (π_s = 1.92) with an adiabatic efficiency of 84.2% at the design point. The mass flow rate is corrected to the standard reference conditions, namely, the standard sea level pressure and temperature. The corrected mass flow rate is defined as

(8.165)

where

(8.166)

(8.167)

The reference pressure and temperature are the standard sea level conditions, namely, p_ref = 1.01 bar (or 101.33 kPa) and T_ref = 288.2 K.

The corrected mass flow rate in a compressor is a pure function of the axial Mach number operating at the standard pressure and temperature. We can easily demonstrate this by writing the continuity equation of one-dimensional flow in terms of total pressure and temperature, the axial Mach number M_z, and the flow area A, namely,

(8.168)

(8.169)

Now, we may multiply both sides by the standard pressure and divide by the square root of the standard temperature to get the “corrected mass flow rate, ”

(8.170)

The performance of a compressor depends on the axial and blade tangential Mach numbers M_z and M_T as discussed earlier. The corrected mass flow rate is a unique function of the axial Mach number and in addition it represents the mass flow rate into the compressor at the standard day pressure and temperature. By defining a corrected mass flow rate, we have basically taken out the effect of nonstandard atmospheric conditions of engine static testing or flight operation. The blade tangential Mach number M_T is proportional to the shaft rotational speed (or angular frequency N) divided by the local speed of sound or the square root of static temperature, namely,

(8.171)

The RHS of Equation 8.171 is called the corrected shaft speed,

The corrected shaft speed is a unique function of the blade tangential Mach number, which along with the axial Mach number determines the performance of a compressor or fan, that is,

(8.172)

(8.173)

It is also customary to nondimensionalize the corrected mass flow rate by the “design mass flow rate, ” and the corrected shaft speed is nondimensionalized by the “design shaft speed” in the compressor performance map. An example of this is shown in Figure 8.53 (from Cumpsty, 1997).

8.8 Compressor Instability – Stall and Surge

The phenomenon of stall in a compressor has its roots in the aerodynamics of lifting surfaces in a high angle of attack. Therefore, from aerodynamics we understand stall as flow separation from the suction or pressure surfaces of an airfoil with large positive and negative angles of attack, respectively. In a compressor operating at a constant rotational (shaft) speed when the mass flow rate drops, the axial velocity decreases, hence the incidence angle increases. We remember from cascade data a parameter that was called “optimum incidence” angle, at which the blade losses were at a minimum. Large deviation from this minimum-loss incidence angle causes a rapid rise in total pressure loss, which signifies boundary layer separation. The positive and negative stall boundaries were then defined for a cascade of a given blade profile shape, solidity, stagger angle, and Mach number. Therefore, when the flow rate in a compressor drops while operating at constant shaft speed, the danger of stall lurks in the background. A stalled compressor flow is unsteady and, hence, offers a means of driving the naturally occurring blade vibrations into resonance. This is the mechanism for energy flow from the fluid into the blade vibration that causes flutter. Flutter is the self-excited aeroelastic instability of an aerodynamic lifting surface. In contrast, buffet (or buffeting) is a forced vibration of an aerodynamic surface that is in the turbulent wake of an upstream wing/fuselage, for example, horizontal tail buffet. Figure 8.53 shows schematic drawing of a compressor rotor with two velocity triangles, one representing its design point operation and the second corresponding to a stalled flow.

The stalled flow in a compressor rotor may be initiated with a single blade at a certain radius. The stalled blade passage acts as a “blocker” and diverts the flow to the neighboring (as yet unstalled) blades. Following the drawing in Figure 8.53b, we note that the flow diversion from the stalled passage to the one above the stalled blade causes an increase in that blades incidence angle, pushing it toward stall. Similarly, the blade below the stalled passage gets a diverted flow, which causes a reduction of the flow incidence angle to the blade below. Therefore, it moves away from stall. We note that the stalled flow in a passage is moving in the opposite direction to the blade rotation, hence it is given the name rotating stall. The angular speed of rotating stall propagation is ∼1/2 of rotor angular speed, that is, ω/2, and in the opposite direction to the rotor rotation. Hence, in the laboratory (i.e., absolute) frame of reference, a rotating stall patch spins with the rotor (i.e., the same direction as the rotor) but at half the speed. To get a feel for the frequency of the rotating stall cells, for example, in a large turbofan engine with a shaft speed of ∼50 Hz, a single stall cell has half the frequency or 25 Hz. Now for a two-to-four stall cells circumferentially arranged around the compressor rotor, the stall cell frequency becomes 50–100 Hz. The rotating stall starts with a single cell and develops into a number of cells with a partial span extent (known as the part-span stall) and may grow into a full-span stalled flow with subsequent reduction in the mass flow rate. This behavior is shown schematically in Figure 8.54. The first appearance of stall is limited to a single cell with subsequent reduction of the flow the average pressure rise drops and the cells multiply into a periodic arrangement, as shown in Figure 8.54. The recovery from the stalled operation in a compressor exhibits a hysteresis behavior, which accompanies systems governed by nonlinear dynamics.

The coupling between the compressor stall instability and the combustion chamber resonant characteristics could lead to an overall breakdown of the flow, or flow oscillation, in the compressor and combustor. The overall flow breakdown in the “system” composed of compressor and combustor is called surge. In a fully developed stage, surge is an axisymmetric oscillation of the flow with a characteristic timescale dictated by the plenum chamber time to empty and fill. In the initial transient stage, surge is asymmetric and thus creates transverse loads on the blades, which may rub on the casing and lead to structural damage and system breakdown. Early contributions to understanding compressor stall and surge and their distinction were made in the United States at NACA (e.g., Bullock and Finger, 1951) and by Emmons, Pearson, and Grant at Harvard (1955).

A unifying approach to treat the compression system instability was first presented by Greitzer (1976). Greitzer developed a successful one-dimensional theory for the onset of surge in a compressor coupled to a plenum chamber cavity, which may serve as a model of the combustor in a gas turbine engine. The elements of the compression system model are shown in Figure 8.55 to be composed of an inlet duct, a compressor, an exit duct connecting to a plenum chamber followed by a throttle that sets the backpressure, and the mass flow rate through the compressor.

Greitzer proposes the ratio of two characteristic timescales as the parameter that governs the dynamics of this system. One characteristic timescale is the compressor throughflow, which may be written as

(8.174)

The second characteristic timescale is that of the plenum chamber, which is the time to charge the plenum chamber to a critical pressure rise condition for a stable compressor operation Δp_c,

(8.175)

Expressing the pressure rise in the plenum as the square of the wheel speed,

(8.176)

(8.177)

and the temperature as the speed of sound squared, Greitzer has shown that the ratio of the two timescales is the square of a parameter B that dictates the fate of disturbances in a compressor, that is,

(8.178)

The critical value of the B-parameter, which causes compressor instability to grow into a surge instead of a localized rotating stall, is shown to be ∼0.7–0.8 by Greitzer. Experimental investigations of compression system instability have supported the proposed model. Figure 8.56 (from Greitzer, 1976) shows the computed results of a transient behavior of a compressor instability that settles into a rotating stall for low B-parameter and surge instability for a B-parameter of 0.6 and 1.58, respectively.

Current understanding of the compressor system instability has allowed strategies to be developed in its active control. Fast detection and cancellation of early oscillations are the keys to a successful active control system design. A review article by Paduano, Greitzer, and Epstein (2001) provides a rich exposition to the subject.

8.9 Multistage Compressors and Their Operating Line

The throttle in a gas turbine engine is the fuel flow control to the combustor. Constant throttle lines are, therefore, the lines of constant T_t4/T_t2. The path of steady-state operation of the compressor with the throttle setting is called the operating or working line in a compressor. It is often superimposed on the compressor performance map, as shown schematically in Figure 8.57. The transient behavior of “spool up” is shown in Figure 8.57 as well. In the context of the stall margin defined as the percentage stall pressure rise divided by the corresponding percentage drop in the mass flow rate, the transient engine operation deviates from the steady-state operating line and approaches the stall or surge line. The level and type of inlet flow distortion also affects the stall margin. We shall discuss inlet distortion and compressor stall later in this chapter.

Compressor performance map for the GE Energy Efficient Engine (E³) is shown in Figure 8.58 (from Cumpsty, 1997). Compressor test rig data as well as the engine test data are superimposed for comparison. Engine measurements are limited to the data points along the engine working line, whereas the compressor rig offers the versatility to produce the entire map. As a point of reference, E³ program was initiated by NASA in 1973 (sparked by the oil embargo) with the goal of 12% reduction in specific fuel consumption over then current commercial aircraft engines of the day. Both GE and Pratt & Whitney developed technology demonstrator engines under this program that exceeded NASA’s goal. A book by Garvin (1998) presents a history of aircraft engine development in the United States with a particular emphasis on GE’s achievements in the commercial engine market. The lead taken by the military needs and the lag of the commercial side in aircraft engine development are presented in the context of the cold war, and the impact of business and strategic teaming/partnerships that have led to a business success. Propulsion engineers need to look at the business side of engineering to appreciate issues of cost, markets, customer service, and product support. For additional reading on the history of aircraft gas turbine engine development in the United States, the book by James St. Peter (1999) is recommended.

Let us do a simple and quick calculation of this compressor’s polytropic efficiency e_c based on its design point pressure ratio and adiabatic efficiency. We estimate the compressor adiabatic efficiency of ∼0.86 from Figure 8.59 at the design point and for the compressor pressure ratio of 23 we substitute these in the modified form of Equation 4.22:

to get an e_c ≈ 0.907. An average stage pressure ratio in 10 stages is

This represents just an average of the stage pressure ratios and we recognize that the front stages operate at a higher loading (front stage pressure ratio of ∼1.6) than the aft stages (pressure ratio ∼1.15) due to a higher blade tangential Mach number M_T.

Now, we shall demonstrate that a throttle position T_t4/T_t2 establishes the compressor pressure ratio, the mass flow rate, and the corresponding shaft speed. From the chapter on cycle analysis and off-design considerations, we argued that the turbine nozzle and the exhaust nozzle throat remain choked over a wide operating range of the engine. As we recall the turbine expansion parameter,

The compressor–turbine power balance demands

To simplify the equation, assume the gas is calorically perfect and neglect the small contribution of fuel-to-air ratio in favor of 1 on the RHS of the above equation, to get

Therefore, the compressor temperature ratio is related to the throttle setting and a constant turbine expansion parameter τ_t via

(8.179)

Here, we see that the compressor temperature ratio increases linearly with the throttle setting. The compressor pressure ratio is related to the temperature ratio via the polytropic efficiency; therefore, the compressor pressure ratio is established by the throttle setting as

(8.180)

The continuity equation at stations 2 and 4, that is, the compressor face and the combustor/ turbine nozzle exit connects the axial Mach number at the engine face to the throttle setting according to

(8.181)

By neglecting the combustor total pressure loss, we may replace the combustor exit total pressure p_t4 with the compressor exit pressure p_t3 to rewrite Equation 8.181 as

(8.182)

We assumed the flow areas A₂ and A₄ remain constant in deriving Equation 8.182 as well as all the gas property variations included in the proportionality constant on the RHS of Equation 8.182. The left-hand side (LHS) of Equation 8.182 is proportional to the corrected mass flow rate , hence

(8.183)

Therefore, constant throttle lines, T_t4/T_t2 = constant, are straight lines on the compressor performance map π_c versus , as shown schematically in Figure 8.59. Also higher throttle constants attain higher slopes, as shown in Figure 8.59, which get the compressor operating point closer to the compressor surge line. Figure 8.59 shows the constant throttle lines and their convergence on compressor pressure ratio of 1, corresponding to a mass flow rate of zero.

The engine face axial Mach number is established by the throttle setting via Equation 8.182 and the compressor pressure ratio was related to the throttle setting via Equation 8.180, therefore,

(8.184)

We may solve Equation 8.184 for the engine face axial Mach number for a given throttle setting. We may also simplify the LHS somewhat by recognizing that the second term in the parenthesis involving axial Mach number is small compared to one, hence applying binomial expansion yields,

(8.185)

Now, let us demonstrate the dependence of wheel speed on the throttle setting. The compressor power is consumed by the rotor blades that experience a torque τ_r and spin at the angular rate of ω. The stator blades do not contribute to the power transfer. We need to sum overall the rotor blade rows in a multistage compressor to calculate the compressor power, namely,

(8.186)

where N is the number of stages. The rotor torque is the integral of angular momentum increase across the blade row and may be expressed as

(8.187)

The geometrical parameters are sketched in Figure 8.60. In their simplest form the integrals are expressed as an average angular momentum at the pitchline radius r_m namely,

(8.188)

The rotor torque, however, is a function of the shaft speed as the absolute swirl downstream of the rotor changes with shaft speed according to

(8.189)

The rotor relative exit flow angle β₂ remains nearly constant with the shaft speed over a wide operating range of the compressor, that is, for attached boundary layer flow to the blades. Substituting Equation 8.189 into Equation 8.188, we get

(8.190)

Expressing the shaft power in terms of the total enthalpy rise and the mass flow rate, we get

(8.191a)

Figure 8.61 shows the parameters in Equation 8.191a.

By substituting for the rotor torque from Equation 8.190 to Equation 8.191a, for the compressor temperature ratio from Equation 8.179, and canceling the mass flow rate, we get

(8.191b)

This is a quadratic equation in ω in terms of the throttle setting T_t4/T_t2, flow angles α₁ and β₂ that remain nearly constant, axial velocities across the rotor, and the radial position of the pitchline stream surface. In practice at a given throttle setting, the shaft finds its own speed consistent with the compressor power consumption and mass flow rate through the machine. We have thus established the interdependence of compressor pressure ratio, corrected mass flow rate, and shaft speed on the throttle setting.

The simplifications introduced in the above discussions may be removed one by one, if we are willing to iterate to find a consistent set of compressor performance parameters. For example, the choking condition in stations 4 and 8 may be relaxed in the case that the nozzle pressure ratio is below critical (i.e., necessary for choking). The subsonic throat Mach number may be found through application of continuity equation using a trial and error approach.

8.10 Multistage Compressor Stalling Pressure Rise and Stall Margin

Koch (1981) developed an analogy between the stalling pressure rise capability of an axial-flow compressor stage and two-dimensional diffusers. Classical diffuser data of Reneau, Johnston, and Kline (1966) and Sovran and Klomp (1967) for a straight centerline, 2D diffuser serve as the point of analogy. The pitchline radius of the compressor stage is taken as the reference point in developing the stall analogy with the diffuser. This approach is useful in the preliminary design phase of an axial-flow compressor in estimating the maximum pressure rise potential as well as the stall margin for a given design point operation. Figure 8.62 shows the diffuser performance data of Reneau, Johnston, and Kline for an inlet boundary layer blockage of 5%. The length of the diffuser is N, which is analogous to L, as shown on the definition sketch on the right (from Wisler, 2000), to represent a diffusion length scale of a compressor blade passage at the pitchline radius.

As a first approximation, the diffusion length L is represented by the arc length of the mean camber line of the airfoil at the pitchline. Assuming a circular arc for the mean camber line, the length L is related to the camber angle ϕ and the chord length c via

(8.192)

Although the area ratio of a diffuser, A₂/A₁, is fixed by the geometry of the diffuser, its counterpart in a cascade depends on the blade incidence, that is, the staggered spacing of the streamtube that enters and exits the blade row. The exit flow area of the blade channel is, however, fixed over a wide operating range of the compressor flow and, therefore, it serves as the reference area in the correlation development by Koch. The inlet flow area is a function of the operating point (i.e., the throttle setting) and decreases with an increasing incidence angle. Figure 8.63 shows the staggered spacing g₁ and g₂ and the flow areas at the inlet and outlet of a compressor blade row.

Here, we have adopted the terminology of Koch in representing the rotor relative velocity as V′ and the absolute velocity as V. A nondimensional diffusion parameter is the ratio of the diffusion length L to the staggered spacing at the exit of the blade row, that is, L/g₂,

(8.193)

where the camber angle ϕ is in radians and the blade inlet solidity is represented by σ₁ at the pitchline. The blade solidity, the camber angle, and the exit flow angle are all reflected in the nondimensional diffusion length ratio in a compressor blade row. The stator blade row diffusion length is calculated in a similar manner using the stator inlet solidity at the pitchline, camber angle, area ratios, and the exit absolute flow angle. The effect of higher camber angle is seen as an increase in diffusion, hence static pressure rise. The effects of blade aspect ratio, tip clearance gap, and Reynolds number are accounted for through the end wall boundary layer thickness. In a diffuser, the inlet boundary layer blockage is a key parameter in the performance and stalling characteristics of diffusers, and hence the end wall boundary layer thickness in a compressor stage serves the same principle.

A stage average approach is adopted in developing a correlation between the stalling pressure rise of a diffuser and a compressor stage. For example, the diffusion length ratio L/g₂ of the stage is the weighted average of the rotor and the stator values with blade row inlet dynamic head used as the weighting factor, that is,

(8.194)

where q′₁ is the rotor inlet dynamic head (relative) and q₁ is the stator inlet dynamic head. Koch (1981) defines an enthalpy equivalent of the static pressure rise in a compressor stage according to

(8.195)

The numerator of Equation 8.195 is the stage static enthalpy rise based on the isentropic stage temperature ratio, which is corrected for the radial shift across the rotor at the pitchline radius. To be comparable, the rotor free work contribution associated with the radial shift needs to be corrected for in the pressure rise comparisons of diffusers and compressor stages. The denominator is the sum of the free stream dynamic heads to the rotor and stator. There are two dynamic heads in the denominator, corresponding to two static enthalpy rises across the rotor and stator in the numerator.

The contributions of Reynolds number, tip clearance, and the axial spacing between the rotor and the stator rows to the stalling pressure rise coefficient are represented in Figures 8.64 to 8.66, respectively (from Koch, 1981). Normalizing values for Reynolds number, ratio of tip clearance to gap, and the ratio of axial distance to the blade spacing are used in presenting the stalling pressure rise coefficient. These are Reynolds number = 130, 000, the average tip (radial) clearance-to-gap ratio /g = 0.055, and the ratio of axial clearance to the blade spacing Δz/s = 0.38. The Reynolds number is calculated based on the blade row inlet relative velocity and the chord length at the pitchline radius. The gap g and the blade spacing s represent a stage average at the pitchline radius in Figures 8.65 and 8.66.

Note that a higher stalling pressure rise is achieved in a compressor stage with a decreasing tip clearance as well as a decreasing axial spacing between the blade rows. The effect of blade stagger was significant on the stalling pressure rise capability of axial-flow compressor stages according to Koch (1981). The effect of stagger is related to a recovery potential of a total pressure deficit region (e.g., upstream wake) interacting with the stator blade row. Leroy Smith first identified this phenomenon in 1958. The velocity vector diagrams upstream of the stator are shown in Figure 8.67. Note that the minimum wake velocity as seen by the stator occurs when the angle between the relative velocity vector V′ and the wake velocity vector is 90°. In case of zero wake velocity in the relative frame, the wake as seen by the stator is equal to U and is in the tangential direction. This point is shown as V₀ = U in Figure 8.67.

The effect of low stagger (i.e., high flow coefficient) is seen in the small included angle between the relative and absolute velocity vectors α + β, which will result in a low momentum wake velocity V_min V. With a low dynamic head associated with this flow, the low stagger stages seem more prone to stall than a high stagger counterpart. In a high stagger stage, that is, a low flow coefficient case, α + β > 90°; therefore, the upstream wake enters the stator with V_wake > V, hence a higher dynamic pressure fluid is less prone to stall. The observations of α + β and 90° and their impact on the loss recovery were made by Ashby in a discussion of Smith’s (1958) paper. Here, we are alerted to the importance of the vector diagram in a compressor stage and its impact on its stall margin. Koch devised an effective dynamic pressure factor F_ef that accounts for the rotor wake interaction with the stator, that is, the effect of stagger on pressure recovery. The formulation of F_ef is presented by Koch to be

(8.196)

The minimum dynamic head is related to the included angle (α + β) of the stage vector diagram according to

(8.197a)

(8.197b)

(8.197c)

The procedure to account for the effects of Reynolds number, tip clearance, axial spacing, and stagger on stalling static pressure rise coefficient in a compressor stage is to first adjust the stalling pressure rise for Reynolds number other than 130, 000 (via Figure 8.63), the tip clearance (to gap ratio) other than 5.5% (via Figure 8.64), and the axial spacing (to blade spacing) other than 0.38 (via Figure 8.65). Therefore, we first calculate the (C_h)_adj. The effect of stagger and the velocity vector diagram is now introduced through the effective dynamic head factor F_ef, which is multiplied by the free stream dynamic head of the stator, that is,

(8.198)

Koch’s stalling effective pressure rise correlation for low- and high-speed axial-flow compressor stages is shown in Figures 8.68 and 8.69, respectively. The maximum recovery of the two-dimensional diffuser for an inlet blockage of 9% of Sovran and Klomp (1967) marks the upper limit in Figures 8.68 and 8.69, which serves as an approximate stalling boundary for axial-flow compressor stages. We note that the stall margin correlation for the low-speed compressor predicts typically below 0.5 of static pressure recovery at stall. The high-speed compressor stages produce a higher effective stalling static pressure rise coefficient of up to ∼0.55. The case of a tandem rotor compressor with an L/g₂ of 2.33 poses the highest stalling static pressure recovery, that is, ∼23% above the diffuser stall curve, as shown in Figure 8.69. The tandem blading serves as a means of flow control in a compressor rotor and the simple diffuser performance underpredicts the stall behavior of a compressor stage that employs flow control. An analogy of a tandem rotor compressor stage should be made with a wide-angle diffuser that employs splitter plates as a means of high static pressure recovery.

Koch demonstrates that effect of the stage reaction on the stalling pressure rise in axial-flow compressor is well predicted by the diffuser stall margin correlation of Figure 8.68 or 8.69. The range of stage reactions from 0.39 to 1.09 (where the stator accelerates the flow, like a turbine) is shown in Figure 8.70. It is surprising that over such a wide range of stage reaction, the stalling pressure rise in a compressor correlates with the diffuser predictions.

In a multistage high-speed compressor, the stagewise distribution of the static pressure rise at stall is compared with the stall margin correlation of Figure 8.71 (from Koch).

Figure 8.71 shows that in a multistage compressor, there is at least one or a group of stages that rises to the predicted level of stalling pressure according to the diffuser correlation. For example, the 10-stage research compressor operating at 95% speed at stall had its eighth stage reach/exceed the stall boundary as predicted by the correlation of Figure 8.69. The 16-stage TF39, at 97% speed, had stages 12 and 15 exceed the stall boundary and stages 13 and 14 operate within 96 percentile of the stall boundary. Note that we are not looking for a perfect match between the correlation and the stall pressure rise of every stage in a multistage compressor. We are rather interested in the trends and behavior of individual and groups of stages and their stall margin. In this context, the shaded symbols in Figure 8.71 represent the “critical” stages, which lie above 95 percentile of the stall boundary. These stages are likely to stall first and cause a breakdown of the flow in the rest of the compressor. The stalling pressure rise data in Figure 8.71 represent actual high-speed multistage compressor tests, which validate Koch’s semiempirical correlation presented in this section. The stall margin correlation based on the diffuser analogy has proven to be a valuable tool in the preliminary design stage of axial-flow compressors.

We have taken the stalling effective pressure rise coefficient of Figures 8.68 and 8.69 and graphed a family of curves with 95%, 90%, …, 70% of the stall correlation (Figure 8.72). We have thus created a chart with 5% stall margin increments as a function of the stage-averaged diffusion length ratio L/g₂. We note that increasing the diffusion length ratio L/g₂, which points to either higher solidity or lower aspect ratio blading, helps with the stall margin. Wide chord blades also operate at a higher Reynolds number, which is beneficial to boundary layer stability, as evidenced in Figure 8.64. The minimum operational tip clearance of ∼1% blade height and the minimum axial spacing between blade rows of ∼0.25 axial chord also create higher static pressure rise in a compressor stage. Stage vector diagram affects the loss recovery (α + β ≥ 90°) and thus stall static pressure rise.