### 13.2.5. Streamline Curvature Throughflow Theory

Vista TF is a general purpose streamline curvature throughflow program for the analysis of all types of turbomachinery, but with special emphasis on single stage centrifugal machines, such as radial pumps, turbines, and compressors, which can usually not be calculated by other throughflow programs.

The program has the ability to compute radial blade rows in both compressors and turbines, and the swirling flows in the radial channels of such machines. This is possible because the blade and channel geometry is defined in a very general way independent of the meridional streamline direction.

This section of the documentation is intended for readers who are not familiar with throughflow methods; it provides a general introduction to the streamline curvature axisymmetric meridional throughflow calculation method. Some details are given in Casey and Robinson (2008) [5]. This section includes a list of technical papers on the subject.

The following topics are discussed:

- The Equations
- The Mean Stream Surface
- The Grid
- Ductflow and Throughflow
- Iterative Solution Procedure
- Initial Estimate
- Radial Equilibrium Equation
- Combination of Velocity Gradient and Continuity Equations
- Relaxation Factors
- Streamline curvature
- Equations for Enthalpy and Angular Momentum
- Boundary Conditions
- Empirical Data
- Blade-to-blade Solution
- Spanwise Mixing
- Streamline Curvature Throughflow Theory: Bibliography

#### 13.2.5.1. The Equations

The equations that are solved are:

- The continuity equation

- The energy equation (Euler equation of turbomachinery)

- A suitable equation of state

- The inviscid momentum equation for the flow on the mean axisymmetric stream surface

Although these equations are inviscid and do not include frictional forces, the effect of the losses are included by empirical changes to the entropy in the equation of state, such that the final solution has a density and pressure field consistent with the presence of losses in the flow.

#### 13.2.5.2. The Mean Stream Surface

This blade-like surface may be regarded as being obtained by averaging all flow properties in the circumferential direction, or as being the flow on a mean stream surface between the blade rows, whose orientation is roughly determined by the blade camber surface. The flow blockage caused by the blades is roughly fixed by the circumferential blade thickness of the blades. It is often known as the S2 meridional stream surface, in contrast to the S1 blade-to-blade surface, and in fact corresponds to the so-called S2m surface in the theory of Wu (1952) [22]. See Figure 13.1: S1 and S2 Stream Surfaces in the Theory of Wu (1952). Note that it has an arbitrary orientation in the circumferential direction and that the flow in the mean stream surface is determined by the slope angles of the surface to the axial and the radial directions, both of which need to be specified in the geometrical input data.

Figure 13.1: S1 and S2 Stream Surfaces in the Theory of Wu (1952)

#### 13.2.5.3. The Grid

The grid for the calculation is based on the streamlines of the mean circumferentially averaged flow in the meridional direction, and fixed calculating stations which are roughly normal to the streamlines. The meridional streamline grid is not fixed, apart from the hub and shroud streamlines on the annulus walls, but changes continually during the iterations so that location of the final streamlines is a result of the solution. The fixed calculating stations are orientated with the blade row leading and trailing edges and are often known as quasi-orthogonals because they are normally nearly orthogonal to the streamlines. The quasi-orthogonals can be in duct regions, that is in the blade-free space upstream and downstream of blade rows, at the leading and trailing edges of the blade rows (which actually are also generally considered to be in the duct region) and internally within the blade rows. See Figure 13.2: Quasi-orthogonal Calculating Stations in an Axial Compressor Stage Calculation. The computational results are available only at the points where the streamlines cross the quasi-orthogonals. In comparison with modern CFD methods, a very crude grid is used with typically 9 to 17 streamlines and 3 to 15 quasi-orthogonals per blade row, giving 50 to 300 nodes per blade row in comparison to 100 000 to 500 000 in modern CFD simulations.

Figure 13.2: Quasi-orthogonal Calculating Stations in an Axial Compressor Stage Calculation

In many early streamline curvature programs, the quasi-orthogonals were generally straight radial lines from hub to casing placed roughly at the location of the leading and trailing edges, but given the more extensive use of lean and sweep in modern designs, and the natural leading edge and trailing edge curvature arising from the variable stagger across the span in most rotors, curved quasi-orthogonals of arbitrary orientation are more appropriate. The use of a grid involving meridional streamlines, all of which proceed smoothly from the inlet plane to the outlet plane of the calculation domain, assumes that there is no reverse flow, and thus operation points with reverse flow in the mean meridional flow (even at the design point) cannot be calculated by this method.

#### 13.2.5.4. Ductflow and Throughflow

Earlier methods, which did not include blade internal calculating planes, are often known as duct flow methods because only planes in the duct regions between the blades, and at the leading and trailing edges, are used. Throughflow refers to methods in which internal planes within each blade row are included as shown in Figure 13.2: Quasi-orthogonal Calculating Stations in an Axial Compressor Stage Calculation. In throughflow methods, the component of the blade force acting along the direction of the quasi-orthogonal can be included in the meridional throughflow equations, so that some global aspects of the effect of the blade lean on the meridional flow can be included. This is not possible in duct flow methods without some empirical adjustment of the pressure gradients.

#### 13.2.5.5. Iterative Solution Procedure

The solution method is iterative in terms of several variables (primarily the meridional velocity, but also the density, streamline location, and other variables), all of which progressively converge to a final converged solution within a fairly complex structure of three nested iterations. In this convergence procedure, many variables lag behind the main iteration for changes in the meridional velocity, so that during one iteration leading to an update of the meridional velocity, many parameters are treated as constant and are then updated based on the new estimate of the meridional velocity prior to the next iteration. It is assumed that this has no effect on the solution provided that a converged solution for all variables is reached.

#### 13.2.5.6. Initial Estimate

The initial solution is generally composed of a first guess of streamline positions (typically by dividing the flow path into equal areas at each quasi-orthogonal) and flow variables at all the grid points (that is, the junction points between the variable streamlines and the fixed quasi-orthogonals). This is then successively refined as a result of each iteration. Once a first estimate of the streamline positions is in place then various terms in the radial equilibrium equation, such as the curvature of the streamlines or the rate of change of meridional velocity along the streamlines, can be determined.

#### 13.2.5.7. Radial Equilibrium Equation

The momentum equation on the mean stream surface is a form of the radial equilibrium equation, giving a velocity gradient equation for the gradient of the meridional velocity along a calculating station, or quasi-orthogonal:

This equation is given in many text books (see Cumpsty (1989) [6]) and is not repeated here. It relates the velocity gradient to the shape and the current positions of the streamlines, to the orientation of the mean stream surface, and to the flow parameters and their gradients from the previous iteration. In its simplest form, this equation is the well-known simple radial equilibrium equation of turbomachinery flows, which gives the relationship between the radial gradient of the axial velocity and the gradient of the swirl in the radial direction, as follows:

In its general form the radial equilibrium equation takes into account flow gradients, streamline curvature, and radial flows, and allows a general arbitrary orientation of the quasi-orthogonal lines. The key difference between the simple radial equilibrium equation and the more general equation is the inclusion of the streamline curvature terms, and it is these terms that give their name to the method. There are several forms of this velocity gradient equation, but the one used here follows the method of Denton (1978) [7], but takes into account the blade force terms as described in chapter 3.4 of the book by Cumpsty (1989) [6].

#### 13.2.5.8. Combination of Velocity Gradient and Continuity Equations

The velocity gradient equation is solved in combination with a method for finding the correct velocity level on the mean streamline that ensures that the velocity along the quasi-orthogonal satisfies the continuity equation giving the correct mass flow across the quasi-orthogonal. See Figure 13.3: Continuity and the Velocity Gradient Used in Determining the Meridional Velocity Level. The meridional velocity on the mean streamline is specified in the innermost iteration, integrated across the flow channel with the help of the meridional velocity gradient, and then continually updated until the mass flow is correct. This requires some care to ensure that the method converges for all mass flows up to the choking mass flow. In this iteration, the density is also needed in the continuity equation. In fact there are two values of the density that can be used, corresponding to supersonic or subsonic flow. In this method, no attempt is made to distinguish these during the innermost iteration; the density is simply taken as constant at the value from the previous iteration, and automatically takes on the supersonic or subsonic value as the iteration converges.

Figure 13.3: Continuity and the Velocity Gradient Used in Determining the Meridional Velocity Level

#### 13.2.5.9. Relaxation Factors

The meridional velocity distribution along the quasi-orthogonal determines the spacing of the streamlines of the meridional flow, and hence the meridional streamline positions. These positions are continually updated as the program converges. Generally, it is not acceptable to use the newly calculated positions of the streamlines directly from one iteration to the next, but a damping factor (often much less than unity) is required to factor the streamline shifts to obtain convergence. A big weakness of streamline curvature methods is that the required damping factor becomes very small when the quasi-orthogonal lines are closely spaced, and so improved accuracy through more calculating planes causes a large increase in calculating time. For this reason, a relatively coarse grid (compared to CFD computations) is used, leading to calculations that take just a few seconds on a modern PC. The problem with closely spaced quasi-orthogonals is related to the large errors in the estimated streamline curvature when a small error in the streamline position is present. It is for this reason that many methods do not include blade internal calculation stations because these automatically decrease the spacing between adjacent quasi-orthogonals.

#### 13.2.5.10. Streamline curvature

The streamline positions can be used to interpolate new blade element data appropriate to their current location and to find the slopes, curvatures, and derivatives of the flow parameters along the streamlines. This data is needed in the radial equilibrium equation (velocity gradient equation; see Radial Equilibrium Equation). The accuracy and stability of streamline curvature methods is related to the prediction of the curvatures of the streamlines. Several different numerical methods for this have been examined. In the current method, the curvatures are calculated with a parabolic approximation through three adjacent points along the streamline.

#### 13.2.5.11. Equations for Enthalpy and Angular Momentum

In the region between blade rows, the total enthalpy and angular momentum may be considered to be convected along the meridional streamlines from the previous station. The entropy is also convected but rises due to the additional losses between the calculating stations. In a blade row, the changes in momentum and enthalpy are calculated from the Euler equation on the assumption that the flow follows the mean stream surface. The mean blade stream surface is roughly oriented in alignment with the camber surface of the blade, but methods are needed for finding the true fluid flow direction taking into account the incidence and deviation of the flow from the mean blade surface. This is generally dealt with using empirical correlations for outlet angle, deviation, or slip factor.

#### 13.2.5.12. Boundary Conditions

At the inlet plane (first quasi-orthogonal) you are generally required to specify the variation of total pressure, total temperature, and angular momentum or flow angle, together with the gas data and information on the mass flow at the outlet plane. For calculations with choked flows, it is not possible to calculate with the mass flow, making it necessary to calculate with specified outlet static pressure, such that the mass flow is a result of the simulation.

#### 13.2.5.13. Empirical Data

Additional empirical methods are used to provide data for the loss production, for the boundary layer blockage, and for the deviation of the flow direction from the mean blade camber surface, so that the effect of viscosity can be taken into account. In fact Denton (1978) [7] argues that “in many applications, throughflow calculations are little more than vehicles for inclusion of empiricism in the form of loss, deviation and blockage correlations, and their accuracy is determined by the accuracy of the correlations rather than the numerics”. The three main effects of the empirical data are:

- In the equation of state, a change in the entropy leads to a pressure loss for a given value of the total enthalpy.

- In the continuity equation, the blockage due to the boundary layer displacement effect leads to a higher value of the meridional velocity.

- In the momentum equation, the deviation of the flow from the blade direction leads to a change in the calculated swirl velocity.

Thus, although the basic inviscid equation of motion used by the method is inherently incapable of predicting entropy rises through the machine, some effects of losses can be included. There are numerous possible combinations of data for the empirical information, based on various definitions of loss coefficients, dissipation coefficients, efficiencies, and so on, and this leads to the largest source of confusion in the data preparation for throughflow programs, and a large and complex array of branching “IF” statements within a typical throughflow program to deal with the alternatives. In making this program, a single form of loss definition has been included based on the entropy rise.

#### 13.2.5.14. Blade-to-blade Solution

The mean stream surface provides the flow field in the meridional plane through the turbomachine. Generally the engineer also needs information on the blade surface velocity and Mach number distributions, so the meridional solution needs to be combined with a blade-to-blade method to find blade surface velocities. This may be either a simple approximation or a more accurate solution of the blade-to-blade flow equations. The more accurate methods enable the exact location of the mean stream surface of the flow to be determined at different blade-to-blade planes, so this can also lead to a further iteration in which the mean stream surface is no longer considered as fixed. Such iterative solutions are often known as S1/S2 solutions (the S1 surface being the blade-to-blade surface and the S2 surface being the meridional surface. See Figure 13.1: S1 and S2 Stream Surfaces in the Theory of Wu (1952)). This technique has generally been replaced these days by fully 3D CFD solutions with mixing planes, but is still of historical interest. The simpler methods generally use a linear approximation for the velocity variation from suction surface to pressure surface. These can be expected to produce sensible results only when a sufficient number of internal blade row calculating stations are included in the grid, probably 5 as the minimum.

#### 13.2.5.15. Spanwise Mixing

A major shortcoming of the basic streamline curvature method is the neglect of the spanwise transport of angular momentum, energy, and losses in the direction normal to the streamlines. By definition, a throughflow program is based on the assumption that the flow remains in concentric streamtubes as it passes through the turbomachine, and no mass transfer occurs across the meridional streamlines which are the streamtube boundaries. In a duct region of a throughflow calculation, enthalpy, angular momentum (swirl), and entropy are taken to be conserved along the meridional streamlines. In reality, there are several mechanisms that lead to an apparent spanwise transport of fluid relative to flow on the mean streamlines, as follows:

- Non-axisymmetric blade-to-blade stream surfaces as a result of streamwise vorticity being shed by the blades (stream-surface twist)

- Secondary flows in the end-wall boundary layers and in the blade boundary layers

- Wake momentum transport downstream of blade rows

- Tip clearance flows with tip clearance vortices

- Turbulent diffusion

As a result of not modeling these effects, throughflow programs in which realistic loss levels are specified for the end-wall regions often predict unrealistic profiles of the loss distribution after several stages because there is no mechanism for the high losses generated near the end walls to be mixed out. The simplest approach to deal with this problem is to specify unrealistic loss distributions across the span that avoid high levels in the end walls. In fact, in preliminary design calculations, this is often adequate. A useful approximation is to specify a mean-line value of loss coefficient or efficiency and to assume that the entropy generated by the losses is the same on each stream-tube. This approximates a complete mixing of the entropy distribution across the span.

Several more sophisticated methods to include more detailed physics of these mixing processes have been attempted so that realistic loss distributions can be specified. The common approach is to model the spanwise mixing as turbulent diffusion, even though some of the effects are due to deterministic flow features. Even with throughflow programs in which spanwise mixing is incorporated, it is still generally not possible to fully incorporate the very high loss levels close to the end walls. Vista TF includes these effects as turbulent diffusion across the streamlines.

#### 13.2.5.16. Streamline Curvature Throughflow Theory: Bibliography

##### Bibliography

[1] G. G. Adkins and L. H. Smith. “Spanwise mixing in axial flow turbomachines”. Trans ASME Journal of Engineering for Power , Vol. 104. pg. 97-110. 1982.

[2] P. M. Came. “Streamline curvature throughflow analysis”. Proc. First European Turbomachinery Conference, VDI Berichte 1185. pg. 291. 1995.

[3] M. V. Casey and O. Hugentobler. “The prediction of the performance of an axial compressor stage with variable stagger stator vanes”. VDI Berichte Nr. 706. pg. 213-227. 1988.

[4] M. V. Casey and P. Roth. “A streamline curvature throughflow method for radial turbocompressors”. I. Mech. E. Conference C57/84. 1984.

[5] M. V. Casey and C. J. Robinson. “A new streamline curvature throughflow code for radial turbomachinery”. ASME TURBOEXPO 2008, ASME GT2008-50187. Berlin . 2008.

[6] N. A. Cumpsty. Compressor aerodynamics. Longman Scientific. New York . 1989.

[7] J. D. Denton. “Throughflow calculations for axial flow turbines”. Trans ASME, Journal of Engineering for Power, Vol 100. 1978.

[8] J. D. Denton and C. Hirsch. “Throughflow calculations in axial turbomachines”. AGARD Advisory Report No. 175 AGARD-AR-175. 1981.

[9] S. J. Gallimore. “Spanwise mixing in multistage axial flow compressors: part II throughflow calculations including mixing”. Trans. ASME, Journal of turbomachinery, Vol. 108. pg. 10-16. 1986.

[10] I. K. Jennions and P. Stow. “The quasi-three-dimensional turbomachinery blade design system, Part I: Throughflow analysis, Part II: Computerized system”. Transactions of the ASME, Journal of Engineering for Gas Turbines and Power Vol. 107. pg. 308-16. 1985.

[11] I. K. Jennions and P. Stow. “The importance of circumferential non-uniformities in a passage averaged quasi-three-dimensional turbomachinery design system”. Transactions of the ASME, Journal of Engineering for Gas Turbines and Power Vol. 108. pg. 240-5. 1986.

[12] K. I. Lewis. “Spanwise transport in axial-flow turbines: Part 2 - Throughflow calculations including spanwise mixing”. Trans. ASME, Journal of turbomachinery, Vol. 116. pg. 187-193. 1984.

[13] B. Liu, S. Chen, and H. F. Martin. “A primary variable throughflow code and its application to last stage reverse flow in LP steam turbine”. Paper: IJPGC2000-15010, Proc. Int. Joint Power Generation Conference, July 23-26. Miami Beach, Florida . 2000.

[14] H. Marsh. “A digital computer program for the through-flow fluid mechanics in an arbitrary turbomachine using a matrix method”. Aeronautical Research Council R and M 3509. 1968.

[15] C. Hirsch and J. D. Denton. “Throughflow calculations in axial turbomachines”. AGARD Advisory report No. 175, AGARD-AR-175. 1981.

[16] K. I. Lewis. “Spanwise transport in axial flow turbines: part 2 - throughflow calculations including spanwise transport”. Trans. of ASME, Journal of Turbomachinery, Vol. 116. pg. 187-193. 1994.

[17] R. A. Novak. “Streamline curvature computing procedures for fluid flow problems”. Trans. of ASME, Journal of Engineering for power, Vol 89. pg. 478-490. 1967.

[18] M. Schobeiri. Turbomachinery Flow Physics and Dynamic Performance. Springer. Berlin . 2005.

[19] L. H. Smith. “The radial equilibrium equation of turbomachinery”. Trans. ASME Journal of Engineering for Power, Vol 88. pg. 1-12. 1966.

[20] D. H. Wilkinson. “Streamline curvature methods for calculating the flow in turbomachines”. Report No W/M(3F). English Electric. Whetstone, England . pg. 1591. 1969.

[21] D. H. Wilkinson. “Stability, convergence, and accuracy of two-dimensional streamline curvature methods using quasi-orthogonals”. paper 35, I.Mech.E. Convention. Glasgow . 1970.

[22] C. H. Wu. “A general theory of three-dimensional flow in subsonic, and supersonic turbomachines of axial, radial and mixed-flow types”. Trans. ASME, Nov. 1952. pg. 1363-1380. 1952.

[23] C. C. Yeoh and J. B. Young. “Non-equilibrium throughflow analyses of low-pressure, wet steam turbines”. ASME J. Eng. for Gas Turbines & Power, Vol 106. pg. 716-724. 1984.

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