A NEW STREAMLINE CURVATURE THROUGHFLOW METHOD FOR RADIAL TURBOMACHINERY

Transcription

1 Proceedings of ASME Turbo Exo 008: Power for Land, Sea and Air GT008 June 9-3, 008, Berlin, Germany GT A NEW STREAMLINE CURVATURE THROUGHFLOW METHOD FOR RADIAL TURBOMACHINERY Michael Casey ITSM - Institute of Thermal Turbomachinery and Machinery Laboratory, Universität Stuttgart, Pfaffenwaldring 6, D Stuttgart, Germany michael.casey@itsm.uni-stuttgart.de Chris Robinson PCA Engineers Limited, Homer House, Sibthor Street, Lincoln LN5 7SB, UK chris.robinson@caeng.co.uk ABSTRACT This aer describes a newly develoed streamline curvature throughflow method for the analysis of radial or mixed flow machines. The code includes curved walls, curved leading and trailing edges, and internal blade row calculating stations. A general method of secifying the emirical data rovides searate treatment of blockage, losses, and deviation. Incomressible and comressible fluids are allowed, including real gases and suersonic relative flow in blade rows. The aer describes some new asects of the code. In articular, a relatively simle numerical model for sanwise mixing is derived, the calculation method for rescribed ressure ratio in comressor bladed rows is described, and the method used to redistribute the flow across the san due to choking is given. Examles are given of the use and validation of the code for many tyes of radial turbomachinery and these show it is an excellent tool for reliminary design. NOMENCLATURE c = absolute flow velocity c d = dissiation coefficient f = mixing factor F d = dissiation force h = secific enthaly i u = unique incidence = indices (calculating lanes and streamlines m = distance along meridional direction m& = mass flow rate M = Mach number N = number of outer iteration o = throat width = static ressure P = arameter (h t, rc u or s q = distance along calculating lane T = temerature r = radius r c = radius of curvature R = gas constant s = secific entroy S = entroy u = blade seed w = relative flow velocity z = axial coordinate Z = number of blades Greek Symbols β = relative flow angle γ = blade lean angle γ in = blending function at inlet γ out = blending function at outlet ε = diffusion coefficient ψ = angle between streamline and lane θ = circumferential coordinate ρ = density Subscrits m = meridional comonent t = total conditions u = circumferential comonent INTRODUCTION In most turbomachinery design systems a meridional throughflow calculation is the backbone of the design rocess. It is fast, reliable, easy to understand, deals easily with multile blade rows and includes emirical loss, deviation and blockage correlations. Performance and exerience from earlier machines can then be taken into account in the Coyright 008 by ASME

2 reliminary design, in a way which is not easy with 3D CFD. Most throughflow codes use the streamline curvature method and derive from those of Smith (966, Novak (967 and Denton (978, based on the general S/S theory of Wu (95. Other methods of solving the equations have been looked at (Simon and Leonard (005, Liu et al. (000, Surr (980, Marsh (968, but none have dislaced streamline curvature in ractice, as in throughflow methods the accuracy is determined by the accuracy of the correlations rather than the numerics (Denton (978. This aer describes a streamline curvature meridional throughflow method for radial turbomachinery known as Vista TF. It is a comletely new coding of streamline curvature throughflow theory based on the method of Denton (978 and its adatation to radial comressors by Casey and Roth (984, but with many new features. The streamline curvature aroach is used to solve the throughflow equations, rather than a more modern numerical technique, as the theory is easy to understand, being based on the sanwise equilibrium of a circumferentially averaged flow in an annulus. It automatically leads to clearly defined meridional streamlines which neatly allow a blade-to-blade and throughflow view of the turbomachine for design uroses. In addition it can be used in a ductflow mode, with only leading and trailing edges of the blade rows defined, which is often an advantage in the reliminary design hase of multistage axial machines. There are two main disadvantages of the streamline curvature technique. Firstly, it allows no reverse flow in the meridional lane. Nowadays, however, issues of flow searation are best resolved with a fully viscous 3D CFD solution rather than with a D throughflow method. It is more imortant that a throughflow code identifies the roblem without breaking down, so that aroriate design decisions can be made to try to avoid the reverse flow. Secondly, the method suffers from a shar increase in calculating time on grids with finely saced quasi-orthogonals owing to the stability requirements of the streamline curvature calculation (Wilkinson (970. However, the calculating time of a throughflow code on a modern lato for a radial stage is only a few seconds, so this is also relatively unimortant. The code described here is rimarily designed for single stage radial turbomachinery alications, but there is no limitation in the code which forbids its use for any multistage axial or radial turbomachinery alication. Key features of the code are listed below: Highly curved annulus walls are allowed roviding a simle definition of axial and radial wall geometries and any combination of these. Any combination of blade row calculating stations, together with duct flow regions, can be used in the domain allowing all tyes of turbomachinery to be calculated. Curved quasi-orthogonal lines allow blades with swee and curved leading and trailing edges to be modeled. Internal blade row calculating stations are used, not ust leading and trailing edges, and blade force terms are included to take into account the lean of the blades, whereby the body force is assumed to act normal to the blade camber surface. A general method of taking into account the sanwise variation of emirical data for losses, deviation and blockage has been rogrammed, including sanwise distributed blockage in the continuity equation and the use of entroy loss coefficients and dissiation coefficients. Dissiation force terms are used in the radial equilibrium equation, although this is mainly of academic interest. Comressible and incomressible fluids are ossible, including suersonic relative flow in blade rows. Blade row choking is not ust included as additional loss, but its effect on the redistribution of the meridional flow distribution is taken into account. In blade rows with sufficient number of internal lanes an aroximation for the blade-to-blade flow field is calculated, which includes the effect of slitter vanes. Sanwise mixing of angular momentum, total enthaly and entroy across the meridional streamtubes is taken into account by a new model which accounts for turbulent diffusion and deterministic secondary flows. The code can oerate with secified mass flow, ressure ratio or secified outlet swirl. A restart from a reviously converged solution reduces the effort for a new calculation with changed geometry or modified flow arameters and boundary conditions, which is useful in combination with otimization methods. Some of these features have either been described in earlier aers on streamline curvature methods, or are relatively straightforward extensions of earlier throughflow methods, and so are not described in detail here. This aer rovides a general introduction to the method used and then concentrates on the comletely new asects of the models in the code and their imlementation. The new features include the way in which losses are taken into account, the use of the code as a mean-line tool, a built-in simlified blade-to-blade model with blending functions for the swirl generation, a new model for sanwise mixing, iteration to ressure ratio for choked comressor blade rows, redistribution of flow due to choked streamlines, and inclusion of different fluids. In addition to describing these new features some details of the validation and verification of the code are given. STREAMLINE CURVATURE THROUGHFLOW Many ublications derive the equations for the streamline curvature throughflow method, so only an overview is given here. The reader who needs more detail should consult two recent books which give a thorough discussion of the method, Cumsty (004 and Schobeiri (005, or refer to the original aers already quoted. The equations solved are the continuity equation, the energy equation (a combination of the first law of thermodynamics and the Euler equation of turbomachinery, a suitable equation of state and the inviscid momentum equation for the flow on the mean stream surface (in the form of the general radial equilibrium equation. The mean stream surface has roughly the form of the blade camber surface and requires geometrical inut and emirical information (incidence and deviation to determine its recise shae. Coyright 008 by ASME

3 The grid for the calculation is based on fixed calculating stations, which are roughly normal to channel walls, and the streamlines of the mean circumferentially-averaged flow in the meridional direction. The meridional streamline grid is not fixed, aart from the hub and shroud streamlines on the annulus walls, but changes continually during the iterations. The fixed calculating stations are oriented with the leading and trailing edges, so need to be curved if these are curved, and can be in duct regions, that is in the blade-free sace ustream and downstream of blades, at the leading and trailing edges of the blades and internally within the blades. By suitable combinations of different tyes of calculating station any tye of turbomachine can be modeled. An examle of the grid for a single stage comressor with radial inlet, centrifugal imeller, vaneless diffuser, crossover bend and return channel with deswirl vanes is shown in figure. This is the first stage of a nine stage multistage radial comressor with 7 blade rows that has been simulated with this method. The grid density shown in figure is tyical of that used, with 5 lanes for a radial imeller. The solution method is iterative in terms of several variables (rimarily the meridional velocity, but also the density, streamline location, etc., all of which rogressively converge from an initial estimate to a final solution within nested iterations. The momentum equation on the mean stream surface is a generalized form of the radial equilibrium equation, develoed to give an exression for the sanwise gradient of the meridional velocity along a calculating station: dcm dht ds d( r cu cm = T + dq dq dq r dq cm ( rcu tan γ + Fd r m The gradient of the meridional velocity is related to the curvature and the current ositions of the streamlines, to the orientation of the mean stream surface (angles ψ and γ and to the flow arameters from the revious iteration. There are several forms of this velocity gradient equation. Equation follows the method of Denton (978, but takes into account the blade force terms as described by Cumsty (004, and the dissiation force terms as given by Horlock (97, essentially as reviously described by Casey and Roth (984. 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 flow across the calculating station satisfies the continuity equation m c + sinψ r c + cosψ c m& = kρcm sinψ dq where k is an emirical blockage factor and ψ is the angle between the streamline direction and the calculating station. The meridional velocity on the mean streamline at each calculating station is secified in the innermost iteration, integrated across the flow channel with the hel of the velocity gradient, equation, and then continually udated until the mass flow is correct. Care is needed in this rocess with m cm + m transonic flows as at M = there is no variation of the mass flow with a change in c m. The meridional velocity distribution determines the osition of the streamlines of the flow on all calculating stations. These ositions are continually udated for each calculating station in an outer iteration as the rogram converges. The streamline ositions are used to interolate new blade element data aroriate to their current location and to find the sloes and curvatures of the streamlines and derivatives of the flow arameters along the streamlines which are needed in the velocity gradient equation, equation. Radial Inlet Crossover Bend Vaneless Diffuser Centrifugal Imeller Return Channel With Deswirl Vanes Fig.. Industrial radial comressor stage showing streamlines and calculating stations and the meridional velocity distribution Between blade rows, the total enthaly and angular momentum are convected along the meridional streamlines from the revious station. The entroy would also be convected in an inviscid flow, but the additional viscous losses causes it to increase in the direction of the flow. In blade rows the changes in momentum and enthaly are calculated from the Euler equation on the assumtion that the flow follows the mean stream surface. The mean stream surface is only roughly aligned with the camber surface of the blade. It oints in the true flow direction taking into account the incidence and deviation of the flow, using emirical correlations. The equation of state is best solved in the form of a Mollier diagram such as =f(h,s, as the enthaly is derived from the Euler equation and the entroy from the losses. For liquids and ideal gases analytical equations are used but for real gases interolation in tables is needed. At the inlet lane the variation of total ressure, total temerature and angular momentum or flow angle together with the gas data are defined. At the outlet lane the mass flow is usually given, but for calculations with choked flows it is necessary to secify the outlet static ressure, such that the mass flow is a result of the simulation. 3 Coyright 008 by ASME

4 EMPIRICAL INFORMATION Emirical methods are used to rovide data for the loss roduction, for the boundary layer blockage and for the deviation of the flow direction from the mean blade camber surface, so that the effect of viscous losses can be taken into account. The three main effects of the emirical data are: In the equation of state a change in the entroy leads to a ressure loss for a given value of the total enthaly. In the continuity equation the blockage due to the endwall boundary layer dislacement thickness leads to a higher value of the meridional velocity. In the momentum equation the deviation of the flow from the blade camber direction changes the mean stream surface and the swirl velocity. There are numerous ossible combinations of data for the emirical information, based on various definitions of loss coefficients, dissiation coefficients, efficiencies and so on, and this leads to the largest source of confusion in the data rearation for any throughflow code, and many internal branches within tyical codes. In this new code the treatment of the blockage, loss and the deviation is searated so that individual correlations for each effect can be alied. Where ossible the correlations for D effects (such as rofile losses and for 3D effects (endwall and clearance effects are also searated. Sanwise variations of each of these can be secified by the user or determined from built-in correlations. Some asects of the deviation model will be described in the next section. Many throughflow methods work with a fixed value of the blockage for all streamlines, such that in the flow is considered to be in a blocked channel. In the resent method, the blockage model includes sanwise distributed blockage, i.e. the value of the blockage factor k in equation can vary across the san. In the solution of the throughflow equations a single form of loss definition based on the entroy rise, following Denton (993 is used, as follows, χ χ turbine comressor = T (s = T (s - s /(½c - s /(½c This entroy-based loss coefficient is shown by Denton (993 to be numerically the same as a kinetic energy loss coefficient. It directly determines the change in entroy which can be immediately used in the equation of state and in equation. In this way it is not only easier to code, but it is also robably a more recise way of including losses in the calculation. Other more common forms of loss coefficient, such as the ressure loss coefficients determined by the many correlations included in the code, first have to be converted internally to an entroy loss coefficient, using the equations given by Brown (97, before they can be used by the code. In addition to loss coefficients or olytroic efficiency, the code can calculate the losses from dissiation coefficients which also directly redict the entroy rise. A single value of the dissiation coefficient for a calculating station is secified, and this value is then used to estimate the total rate of entroy roduction on the wetted surfaces due to the boundary layer dissiation based on the integration of where w is the local surface relative velocity at the edge of the boundary layer. In a duct region the integration is carried out on the hub and casing walls, and in a blade row the integration includes the dissiation on the suction and ressure surfaces using the local relative surface velocities. The total entroy roduction is then used to determine a mean secific entroy rise from one calculating station to the next and this is alied on each streamline. As the code allows for liquids, ideal gases and real gases, some care is needed in the determination of efficiency from the results, and asects of the calculation of efficiency in the code have already been ublished in Casey (007. BLADE-TO-BLADE SOLUTION The solution on the mean stream surface rovides the flow field in the meridional lane through the turbomachine, and this needs to be combined with a blade-to-blade method to find blade surface velocities. In traditional S/S methods this is done with the hel of an additional S blade-to-blade method. The current code includes blade internal calculating stations and, if sufficient of these are resent to calculate a reasonable estimate of the streamwise gradient of swirl, then this can be used to estimate the blade-to-blade loading from the local circumferential blade force, similar to Stanitz and Prian (95, as follows: ( rcu = (ρc θ m The method comutes the flow based on the geometry of the mean stream surface. This is not congruent with the mean camber surface, and the differences (due to incidence and deviation have to be taken into account by emirical modifications. This is done with blending functions which adat the swirl generation in the blade row to allow the camber surface to be artly transarent to the flow, such that the flow angle differs from the blade angle as outlined by Trauel (977 and used by Casey and Roth (984. A similar aroach using so-called dearture angles is described by Smith (00. When calculating radial turbomachinery of high solidity, the blending functions on the swirl are used in the inlet region and the dearture angle aroach is used in the trailing edge region. Extensive tests on many different tyes of blade row have demonstrated that this is the most effective aroach in high solidity blading and leads to sensible estimates for the blade-to-blade loading in the inlet and outlet regions of the blade. The angular momentum of the flow relative to the blade row is calculated as follows: ( rw = ( rw + u ( γ in TS& [( rc tan( β + δ ( γ ][ ( γ ] m total u in te out in The actual value of the swirl at inlet and the deviation angle at outlet (determined by the deviation or sli correlations need to be udated each iteration. The blending functions (γ in and γ out follow the aroach of Wilkinson (969. Figure shows the redicted suction surface and ressure = ρc w da d m 3 4 Coyright 008 by ASME

5 surface static ressure distribution along the mean meridional streamline of a radial imeller, calculated using CFD (ANSYS CFX and the aroximate blade-to-blade method. Although the simlified method is not erfect, it is clearly sufficiently accurate to guide the designer to make sensible design decisions about blade loading distributions. Figure : Suction and ressure surface static ressure distribution along the mean meridional streamline of a radial imeller ith 3D CFD (CFX and with throughflow. SPANWISE MIXING A maor shortcoming of the basic streamline curvature method is the neglect of sanwise transort of angular momentum, energy and losses in the direction normal to the streamlines. By definition, a throughflow code is based on the assumtion that the flow remains in concentric streamtubes as it asses through the turbomachine, and no mass transfer occurs across the meridional streamlines which are the streamtube boundaries. So for examle, in a duct region of a throughflow calculation enthaly, angular momentum (swirl and entroy are conserved along the streamlines. In reality, there are several mechanisms that lead to an aarent sanwise transort of fluid relative to flow on the mean streamlines, as follows: Non-axisymmetric blade-to-blade stream surfaces as result of streamwise vorticity being shed by the blades (streamsurface twist. Secondary flows in the endwall boundary layers and in the blade boundary layers. Wake momentum transort downstream of blade rows. Ti clearance flows with ti clearance vortices. Turbulent diffusion. If realistic loss levels are secified for the end-wall regions, and sanwise mixing is neglected, then unrealistic rofiles of the loss occur after several blade rows as there is no mechanism for the high losses generated near the end-walls to be mixed out. The simlest aroach to deal with this roblem is to secify unrealistic loss distributions across the san which avoid high levels in the end-walls. In fact, in reliminary design calculations it is often adequate to secify a mean-line value of loss and to assume that the entroy generated is the same on each stream-tube. This aroximates a comlete mixing of the losses across the san. More sohisticated methods to include the hysics of these mixing rocesses have been attemted so that realistic loss distributions can be secified. The common aroach is to model the sanwise mixing as a turbulent diffusion, even though some of the effects are due to deterministic flow features. Sanwise mixing is needed to mix out the high losses close to the endwalls, but the recise model of how the mixing is included aears not to be articularly imortant (Denton and Hirsch (98, Adkins and Smith (98, Gallimore (986 and Lewis (994 The model in this code follows that described by Denton (Denton and Hirsch (98. Imrovements to this are guided by the turbulent diffusion model of Lewis (994. The model assumes that some roortion of the local flow is sread across the streamlines by deterministic sanwise flows. In a duct flow, the entroy, angular momentum (swirl and total enthaly on a articular grid oint are determined mainly by the values on the same streamline at the ustream station and artly by the values transferred from the adacent streamlines. A fraction of the flow (-f is convected along the streamlines and a fraction (½f is transferred from each of the two adacent streamlines, where f is a mixing factor with a value less than unity, see figure 3. f f / f / i i i + = ( f Pi, + ( f / Pi + + ( f / Pi + Figure 3: Mixing factor model roosed by Denton (98 The value of the convected arameter P is first calculated on the assumtion of no mixing and this is then modified by a small amount due to mixing as follows: P In this way a fraction f of the conserved arameter P diffuses away from the streamline (actually f/ to the uer streamline and f/ to the lower streamline and a fraction f/ of the values on the uer and the lower streamlines diffuses to this streamline. This can also be written in the following form Pi, = ( f / [( Pi + Pi ( Pi Pi ] If we write the difference between the arameter P on adacent streamlines as δpi + = Pi + Pi and δpi = Pi Pi we obtain 5 Coyright 008 by ASME

6 Pi, = ( f / ( δpi + δpi 3 Note that with a ositive gradient in P along the calculating station there is a ositive contribution to the value from the uer streamline and a negative contribution from the lower streamline due to mixing, and if the gradient is constant then this leads to no change in the arameter P. This is very effective in causing mixing as the flow roceeds downstream, but it has a large drawback as a general model: If the meridional grid sacing or the number of streamlines are changed, then a different value of the mixing factor is needed to roduce the same level of sanwise mixing. Denton suggested simly that a value of f = 0.5 should be used to cure any roblems of entroy buildu in multi-blade-row calculations. This disadvantage can be overcome if a turbulent diffusion equation is used to determine the strength of the mixing factor, as exlained below. Following the aroach of Lewis, we assume that the sanwise mixing of a arameter P, which may be angular momentum, total enthaly, or entroy, is determined by a diffusion equation of the tye P P c m = ε m q where m is the meridional direction, and q is the sanwise direction, P is the arameter undergoing sanwise mixing and ε is a diffusion coefficient. For simlicity in this model q is taken as the distance along the quasi-orthogonal rather than the exact sanwise direction, although this simlification could easily be removed if necessary. For a small ste along the meridional streamline we obtain that the change in the arameter P due to sanwise mixing by diffusion is given by δm P P = ε c m q The second derivative along the calculating station can be written as P P = q q q If the gradient of arameter P is constant then this term is zero and no sanwise mixing takes lace, as the sanwise transfer due to diffusion of P from the streamline with the higher value is comensated by the sanwise transfer from the adacent streamline with the lower value. An aroximate value of this second derivative is P Pi + Pi Pi Pi = q q, +,, +,, i qi i qi qi qi q and if the streamlines are evenly saced, we obtain q = ( q q = ( q q = δq q + i + The second derivative of P can be aroximated by q P δq so that we obtain [( P P ( P P ] i + i i i q P = δq ( δp δp i + i The change in arameter P due to sanwise mixing becomes δm ε Pi, = ( δp,, i + δpi c δq m, This includes a ositive contribution transferred from the uer and the lower streamtubes and a loss due to diffusion to these streamtubes. By comarison with equation 3, it can be seen that this is formally identical to the mixing factor algorithm. This algortihm is equivalent to the solution of the diffusion equation if the mixing factor is directly related to the hysical diffusion coefficient as follows: f / δm = c m, δq The actual value of the mixing factor f is not a constant but needs to be changed throughout the flowfield. More diffusion, wider sacing of the quasi-orthogonals, lower sacing of the streamlines, or a lower value of the meridional velocity all require a higher value of the mixing factor. Equation 4 can also be written as the roduct of two dimensionless terms ε / cm, δq f / = δq δm The numerator reresents the rate of sread of arameter P across the streamlines under the influence of diffusivity. The denominator is related to the grid structure, as it is the tangent of the angle between adacent neighboring oints, and reresents the sread of the grid. In this way it can be seen that the mixing factor needs to be adusted to take into account the sread of arameter P relative to the sread of the grid. This exoses a clear weakness of the method. If the grid has wide sacing along the meridional direction together with small sanwise distances between the streamlines the factor f becomes very large. Clearly it is not sensible if this becomes too large. With a value of the mixing factor of 0.5, as suggested by Denton, the algorithm causes the flow on any articular streamline to have the same influence as that from the adacent streamlines and more mixing than this is not really ossible across a single streamtube. A second weakness is that the algorithm as given above only accounts for changes that take lace across a single streamtube. At high mixing levels or with closely saced streamlines the next adacent streamtubes are also affected by diffusion, so that some of the conserved arameter on these streamtubes is also transferred to the streamline under consideration. If we assume a constant grid sacing from streamline to streamline and examine the contribution to arameter P that is transferred by diffusion from all adacent streamlines we obtain the result that this effect diminishes with the square of the grid sacing, as shown in equation 4 above. Taking into account the effect from all streamtubes gives additional terms in the mixing algorithm so that the ositive contribution from all adacent streamtubes is ε 4 6 Coyright 008 by ASME

7 P + =... + ( f + ( f + ( f / P / 8 P /8 P i + i i ( f + ( f + ( f 3 + / P /8 P / 8 P i i + i 3 The negative contribution that is lost from the local streamline is given by P = ( f + f / 4 + f / 9 + f /6 + L Pi, The maximum value of the infinite series here is π / 6 so that if not more than 00% of the local value can be diffused away this leads to a maximum value of the mixing factor of f = 6/ π 0.6 In the current mixing algorithm the first 4 terms in this series are taken into account so a minimum of 9 streamlines is needed if sanwise mixing is used. The analysis above shows formally that, with the aroriate coefficients and adustment to take into account the effect over several streamtubes, the mixing factor model can be identified with a turbulent diffusion equation and so can be used to model this effect by aroriate tuning of the mixing factor. The mixing factors associated with each individual source of sanwise transort in each streamtube (including turbulent diffusion can be combined to obtain the cumulative effects. In ractice, the code makes use of this algorithm, but uses a mixing factor determined from a usersecified eddy diffusion coefficient from equation 4. In the interest of simlicity, the actual value of the mixing factor is calculated searately for each streamline in the flowfield based on constant values of the diffusion coefficient, and the same value of f is used to redistribute the arameter P uwards and downwards to adacent streamlines. This aroach ensures that the san-wise mass-averaged values of the arameter P on the calculating station are conserved desite the mixing between the streamlines. In addition the algorithm takes into account the non-uniform sacing of the streamlines and the secial streamlines close to the wall where only one adacent streamline is resent. The data given by Gallimore (986 and Lewis (994 identify a hysically realistic value for the diffusion coefficient ε scaled with the mean meridional velocity and the stage length for axial turbines and comressors. There is considerable scatter between different machines and different oerating oints. A larger value is needed where sanwise mixing is high due to deterministic effects (secondary flows, streamsurface twist, etc. and a lower value is required to account for ure turbulent mixing, which is higher in comressors than in turbines. In the current calculations the actual value for the mixing coefficient used is based on the numerical values of Gallimore and Lewis but is scaled by a reference velocity and the reference diameter of the calculation, as follows ε = 0.000u ref D ref 5 In radial turbomachinery the reference condition is imeller outlet for comressors and imeller inlet for turbines. The mixing algorithm has been incororated to redistribute the ressure losses generated by the endwall boundary layers, but an examle of its use for another urose... is shown in figure 4. This shows a simulation of a radial imeller for an ethylene refrigeration alication with a cold sidestream. In the calculation with no mixing the cold (blue and warm (red inlet flows do not mix through the whole comressor and remain stratified to the outlet. Using the sanwise mixing model with a diffusion coefficient given by equation 5, as recommended above, leads to mixing which closely matches that of a CFD simulation of this imeller. Throughflow No Mixing Throughflow Mixing 3D CFD Figure 4: The effect of sanwise mixing on the temerature stratification in a refrigeration imeller with a cold sidestream, Left: No mixing Middle: Mixing (equation 5, Right: 3D CFD CHOKING Before discussing how choked flows are calculated three imortant oints need to be made. Firstly it should be noted that the throughflow method is not articularly well suited for choked blade rows. The mean stream surface equations average out the flow in the circumferential direction and are thus not really aware of any high Mach numbers on the suction surface of blades. In addition, any shocks that may be resent in turbomachinery flows are generally not oriented in the circumferential direction, so they are smeared out in the circumferential averaging of the flow to determine the mean stream surface. Nevertheless, desite these serious limitations an attemt has been made to model choking in the blade rows so that, in combination with correlations, the maximum flow and the additional losses related to shocks are taken into account in the overall redicted erformance. In this way the code includes asects of choking that are comatible with the level of emiricism of tyical D calculation methods. This is useful in a code intended for design uroses, as it identifies choking roblems early in the design rocess, and aids the understanding of the matching of the blade rows as the rotational seed varies. Secondly, in a fully choked flow it is better to calculate with a secified ressure ratio rather than with a secified mass flow. Simulations in which the secified mass flow exceeds the choking mass flow lead to a hysically imossible solution and there are many solutions with different ressure ratios available for the choking mass flow. Denton (978 exlained very briefly how to solve this roblem for turbine blade rows where choking takes lace near the trailing edge 7 Coyright 008 by ASME

8 based on the so-called target ressure method, which is also described in more detail by Came (995. The new code includes these techniques for turbines, but this has now been extended to comressor alications where choking occurs near to the leading edge. Thirdly, choking occurs at the throat between two blades. Although the throat is generally not a calculating lane of the throughflow calculation it can be used as a virtual lane to assess whether the streamline concerned is choked and to limit the mass flow accordingly. A good estimate of the choking flow should make use of accurate estimates of the throat areas and throat osition so the throughflow code needs to be combined with a geometry definition system to ensure that the throats and throat ositions are well-defined. This is not a ractical limitation as the blade geometry data and the channel geometry also have to be reared by such a rogram. The choking calculation can be considered to be a straightforward extension of the simle one-dimensional isentroic flow of a erfect gas through a stream-tube of varying area. Classical one-dimensional gas dynamics then determines the mass flow er unit streamtube area as a function of the Mach number and the maximum mass flow er unit area at a Mach number of unity. In a throughflow calculation we do not have a one-dimensional flow but have a series of individual streamtubes across the san. The choking of each individual streamtube must be analyzed on a onedimensional basis and the maximum ossible mass flow for the calculating station can then be calculated by integrating the maximum mass flow at each streamtube across the san. In this rocess an individual streamtube on a articular station can be choked but others are still able to ass more flow so the whole calculating lane is not yet choked. Close to the leading edge of a comressor and near to the trailing edge of a turbine the maximum value of the local mass flow er unit streamtube width is limited by the throat to be: m& max, th = Zo t γ + RT / γ t γ + ( γ The equation for the maximum flow across the calculating station adacent to the throat is then determined by integration across the san as m& max, th = ( km& max, th sinψ dq At a turbine outlet, the rocedure used is similar to that described by Denton (978 and Came (995, so this will be described first. They assumed that the choking of a turbine always occurs at the throat which is taken to be close to the turbine trailing edge lane. Their calculations were generally for axial blade rows without internal lanes, and they assumed that there are no relative total ressure losses along a streamtube between the turbine leading edge inlet lane and the turbine throat. In the new code with internal blade row calculating stations it is assumed that there are no losses from the next ustream quasi-orthogonal to the throat. This would be the leading edge if no internal lanes are included. In rotors of radial turbines, and in turbines with a high flare, there may be a radius change between the stations and this needs to be taken into account to determine the local relative total ressure and temerature at the throat. These can be determined from the values at the ustream calculating lane on the assumtion of adiabatic isentroic flow and from the condition that the rothaly is conserved in the imeller. For choked turbine outlets the effect of suersonic deviation needs to be included, and in the throughflow code this causes the outlet flow angle in suersonic flow to be determined from the continuity equation rather than from correlations. Choking at a comressor inlet is more comlex as it can occur through three searate mechanisms, see Cumsty (004. Firstly, if the inlet flow is subsonic, choking will occur if the Mach number reaches unity at the throat between the blades. This can occur at subsonic inlet Mach numbers with high negative incidence, or at low incidence with very thick blades with high blade blockage and a small throat area. Both cases give rise to an acceleration from a subsonic flow to a Mach number of unity at the throat, so there are few additional losses caused by this rocess, other than incidence effects. Secondly, if the inlet flow is suersonic then the blade can also choke at the throat. If the flow chokes at the throat this imlies that first there is a detached shock from the suction surface of the blade to ustream of the adacent blade. The suersonic inlet flow becomes subsonic at this shock and then re-accelerates to be suersonic again at the throat. The relative total ressure at the throat is thus lower than that at the inlet because of the losses across the shock, and these losses are generally modeled as if they occur in a normal shock. Thirdly, if the inlet flow is suersonic at higher inlet Mach numbers (say M >. choke may occur due to unique incidence. At unique incidence, the flow in the inlet remains suersonic u to and including the throat. The flow is choked ustream of the throat by the suersonic exansion wave between the leading edge of the uer blade and the suction side. Lower incidences are not ossible than the unique incidence condition, as they would imly a higher mass flow than this choking mass flow. The exact mechanism of this is exlained by Freeman and Cumsty (989 for axial comressors and has been extended to transonic radial comressors by Lohmberg et al. (003. For choking at the throat it is assumed that this is close to the comressor inlet and that there are no changes in relative total ressure and temerature between the inlet lane and the throat, excet those which may occur in the detached shock. It is assumed that the detached shock is normal to the flow and that the shock Mach number is the same as the inlet Mach number to the blade row. If the streamtube is choked at the throat then this limits the maximum mass flow on this streamtube and limits the maximum meridional velocity of the flow leading to a lower limit on the incidence that is ossible. Lower incidences are not ossible as they would imly a higher mass flow than the choking mass flow. In this way this mechanism for choking also roduces a lower limit on the incidence as in the unique incidence condition. At a throat, the value of the mass flow is checked during the inner iteration for mass flow in the iterative rocedure. If the local value is found to be above the maximum value of a 8 Coyright 008 by ASME

9 choked flow, then a limit on the meridional velocity on this streamtube is alied. The choking of an individual streamtube then automatically redistributes the mass flow across the inlet lane of a choked comressor blade row. An examle of this is given in figure 5. This shows the distribution of incidence and meridional velocity across the inlet lane of an industrial radial comressor. The imeller was designed for an axial inlet flow but used in a multistage machine with a radial inlet leading to a severe gradient of the meridional velocity across the san due to the shar inlet curvature. The calculation taking local choking into account automatically limits the mass flow in the outer choked streamtubes so that more flow enters the inner unchoked streamtubes. The same effect can be seen in the distribution of incidence in the outer streamtubes, which cannot reduce below that at choke. Meridional Velocity (m/s % San Choking by unique incidence is also dealt with by alying a limit on the meridional velocity. Assuming a correlation is available for the unique incidence (i u or that this is known from other data, then the maximum flow angle at the leading edge on a articular streamline can be calculated from the blade angle. The swirl velocity ustream of the leading edge is known and this allows the maximum value of the meridional velocity due to unique incidence to be estimated from the blade inlet angle, as follows: ' c = c / tan( β + i m, max u u Cm with Cm without Incidence with Incidence without Figure 5: The effect of the choking model on the flow and incidence distribution at inlet to a choked comressor for a stator and a similar equation with the relative swirl velocity for a rotor. If this value is less than that which would occur due to choking at the throat, then this is used to limit the meridional velocity at the leading edge lane on this streamtube. It should be noted that choking of any articular streamtube means that the meridional velocity of this streamtube is no longer determined by the velocity gradient equation from radial equilibrium theory, but rather by the maximum Incidence ( meridional velocity determined from the continuity equation. In some blade rows calculated with many internal blade calculating lanes, the second and even the third calculating lane may actually be ustream of the effective throat, so an error results from the assumtion that the throat is at the leading edge lane. This is taken into account by secifying the location of the throat together with geometrical throat area as inut data. ITERATION TO PRESSURE RATIO The target ressure method of Denton is used to converge the iteration to a rescribed ressure ratio. In this mode, the oerating oint is defined by the exansion ratio for turbines (or ressure ratio for comressors between the inlet stagnation ressure and the static ressure at the trailing edge of the last blade row on the mid-san streamline. At all other trailing edges a value for the static ressure on the mid-san streamline is estimated, whereby these ressures are called the target ressures. These estimates can be aroximate as they are imroved during the iteration rocedure to be consistent with the mass flow through the machine and the secified overall ressure ratio. During the iterations the normal internal mass flow iteration rocedure is used at all lanes which are not trailing edges, in that the meridional velocity distributions are obtained from equation to satisfy continuity with the current value of the inlet mass flow. At trailing edge lanes a different rocedure is used. Here the meridional velocity on the mid streamline is adusted not to match continuity but to achieve the target mid-san static ressure. The difference between the actual ressure and the target ressure is used to correct the estimate of the meridional velocity on these lanes as follows: target = cos δc m ρc m This can be derived from the Euler equation, using the assumtion that a small change in meridional velocity does not change the losses or the relative flow outlet angle. The corresonding mass flow at the trailing edge is then found by integration of equation. In the first instance, there is no attemt to satisfy continuity with the inlet mass flow. The rogram continues for a maximum number of outer iterations at which oint the target ressures will have been achieved with sufficient accuracy, or fewer if the target ressure has already been achieved. At this oint the estimates of target ressures and the inlet mass flow are revised. The estimates of target ressure are adusted to imrove agreement between the mass flow through the next downstream trailing edge and the current estimate of the inlet mass flow, using the simle correction formula: N N = N m& inlet i i N m& i where N is the number of the outer iteration. The change in target ressure is relaxed to ensure stability. The inlet mass flow is then udated to be that through the first trailing edge lane. The inlet mass flow is also relaxed. In a choked comressor blade row the losses within the β 9 Coyright 008 by ASME

10 bladed region are no longer a function of the mass flow, which is fixed, but become a function of the back ressure. This is rather like the situation in a choked inviscid D convergingdiverging Laval nozzle, where the back ressure determines the location and strength of the shock, and the level of losses that occur are a function of the shock strength. At lower back ressures in a D Laval nozzle the shock moves backwards in the diverging channel and becomes stronger with more losses at the same time. The shock losses in a choked comressor blade row are modeled in a similar way. The equations given above are first used to identify the maximum mass flow at the throat lane. If the flow is choked and the calculation is at a secified ressure ratio, then additional losses need to be generated within the blade row so that the mass flow at outlet (where the target ressure is secified matches the choked mass flow. Without additional losses the outlet density would be too high and the mass flow on the blade trailing edge would be too large. The additional losses are distributed evenly across the san and uniformly downstream of the first calculating lane which is unchoked. In each iteration the additional losses are determined from the condition needed to correct the mass flow at the trailing edge. This algorithm assumes that when the target ressure is achieved (d = 0, then from the Gibbs function we have Tds = dh vd, ds = dh / T = c ( dt / T The equation for an ideal gas can be differentiated to give dρ d dt v = RT, = ρ T and if d = 0 these equations can be combined to give ds = c ( dρ / ρ The error in the density is assumed to be related to the error in the mass flow at the trailing edge so we obtain that the additional losses that are needed to match the mass flow at the trailing edge with the choked mass flow at the inlet can be estimated from the trailing edge mass flow error as s c ( & m / m& The error in the mass flow at the trailing edge is thus used to udate the losses within the blade row until both the mass flow and the target ressure at the trailing edge are correct. In this rocess the change in the additional losses is damed in each iteration. The additional losses in this rocess are determined by the rogram and are in addition to any losses that may be secified by the user or determined by the secified correlations. Few engineers are aware of this feature of choked flow calculations, whereby the level of losses are determined directly from the ressure (or density ratio rather than the detailed aerodynamics of the blading. EQUATION OF STATE Internally the losses are defined via a change in entroy using an entroy loss coefficient, and the total enthaly is determined by the Euler equation, so that the form of the equation of state that is most useful is = f ( h, s, ρ = f ( h, s Many loss coefficients used in turbomachinery correlations are defined not as entroy loss coefficients but as ressure loss coefficients, so the code internally converts these to entroy losses so that the oeration with the equation of state always involves the arameters h and s, and the subroutines involving the equation of state remain relatively simle. The forms of the ideal gas equations used by the throughflow code can be derived by integration of the Gibbs equation for an ideal gas, and lead to the following equations / = γ /( γ ( s s / R ( h / h e /( γ ( s s / R ( ρ / ρ = h / h e The Gibbs equation can also be integrated to find the equation for the ressure in an incomressible calculation, as follows = ρ[( h h c T ( e ( s s / c together with an equation for the density, which is constant. In the case of real gases, the real gas roerties are currently being incororated via tables of values rather than as equations of state, as this rocedure is only needed once for all ossible gases. For reasons of consistency with other codes the real gas data are rovided to the code in the form of tables of roerties in the form of h = f (, T, ρ = f (, T in a standard file format known as.rg files within the ANSYS CFX software system. At each ste in the streamline curvature iteration rocedure the values of h and s are udated from the losses and the Euler equation. The throughflow code then corrects the other gas roerties to find better estimates from the real gas tables. In this rocess the values of * and T* are first taken from revious iteration (here denoted by an asterisk and corrected to an imroved estimate consistent with h and s. First the value of T consistent with the new value of h and earlier value of * is found from table of roerties such that: h = f ( *, T Then a new value of s consistent with T and * is found from s = f ( *, T together with a new value of density (secific volume consistent with T and * v = f ( *, T Finally, the Gibbs equation is used to find a better estimate of the ressure as follows Tds = dh vd, dh = 0, d = Tds / v * = ( T / v ( s s These stes could then be reeated to convergence, but as this rocess is embedded within the streamline curvature iterations, it automatically converges on the correct value when the whole solution has converged. MEANLINE CALCULATION A novel feature of the code is its ability to run as a quasi mean-line method. In this rocess the sanwise velocity gradient (equation can be multilied by a user-secified factor less than unity. If a factor of zero is used, then the code effectively becomes a mean-line code with no sanwise ] 0 Coyright 008 by ASME