Influence of a Favorable Streamwise Pressure Gradient on Laminar Film Cooling at Mach 2.67

Transcription

1 4 TH EUROPEAN CONFERENCE FOR AEROSPACE SCIENCES Influence of a Favorable Streamwise Pressure Gradient on Laminar Film Cooling at Mach 2.67 Michael Keller and Markus J. Kloker Institute of Aerodnamics and Gasdnamics, Universit of Stuttgart Pfaffenwaldring 21, Stuttgart Abstract Direct numerical simulations are used to investigate the method of film cooling in a laminar flat-plate Mach-2.67 boundar laer. Air is emploed as mean-flow and coolant gas and is introduced into the boundar laer through one row of holes b means of a modeled blowing approach. The effects of a favorable streamwise pressure gradient are investigated for two different blowing rates and an adiabatic-wall boundar condition. It is shown that a favorable streamwise pressure gradient has a beneficial influence on the cooling effectiveness, mainl due to the higher streamwise velocit close to the wall. 1. Introduction In order to protect the thermall highl loaded regions of combustion chambers and rocket noles of net-generation space transportation sstems innovative and efficient cooling strategies have to be developed. A promising ansat is the method of film cooling which is a viable cooling technique in high speed flows and perfectl suited for active thermal protection sstems. The influence of film cooling through discrete holes and spanwise slits on laminar super- and hpersonic boundar laers has been investigated etensivel b Linn & Kloker, e.g. [1]-[3]. Direct numerical simulations were used to anale the effects of geometric and gas-dnamic parameters, such as hole spacing, hole arrangement, Mach number, Renolds number, blowing rate, inclination angle, compound angle and wall-temperature condition. From a coolingperformance point of view, slits are better than holes when imposing the same total mass flu, and staggered hole rows are better than aligned rows. Also, a small spanwise spacing is preferable in order to generate a more homogeneous cooling film. It was furthermore demonstrated that the wall boundar temperature isothermal, adiabatic or radiationadiabatic as well as an inclination angle and compound angle strongl affect the cooling performance. Corresponding eperimental studies have been performed b Heufer, Hombsch & Olivier, e.g. [4]-[6]. The investigated the effects of spanwise slits and discrete holes as well as the influence of the blowing ratio, Mach number, Renolds number, various coolant gases, and flow acceleration. A comparison of the numerical results with the eperimental findings for a laminar Mach-2.67 boundar laer with an isothermal wall and spanwise slits showed good agreement [1]. The flow regime in a rocket nole is characteried b a strongl decreasing pressure and a strong flow acceleration. Hence, the influence of a favorable streamwise pressure gradient (50% linear pressure decrease within the integration domain of approimatel 186 δ 0 ) is presented for blowing through one row of holes. Its effect on the developing flow structures and the cooling effectiveness is demonstrated b comparison with a reference case with ero pressure gradient. The remainder of this paper is organied as follows: Chapter 2 describes the governing equations and the numerical procedure, boundar conditions, and simulation setup of the problem. The results are presented in Chapter 3. Finall, Chapter 4 summaries the main findings and contains some concluding remarks. Copright 2011 b M.Keller and M.J.Kloker. Published b the EUCASS association with permission.

2 ADVANCED NUMERICAL METHODS ID-162/ M.KELLER, M.J.KLOKER 2. Numerical Method 2.1 Governing equations The governing equations are the continuit equation, the three-dimensional compressible Navier-Stokes equations and the energ equation, in the following written in dimensionless vector form: where (ρe) t ρ = 0 t, (1) ( ρ v ) + (ρ v v ) 1 + p= and t Re σ (2) 1 + (p+ρe) v= (κ 1) Re Pr Ma 2 (ϑ T)+ 1 ( σ v ), Re (3) ( ( v+ v T σ=µ ) 2 ) ( ) v I 3 describes the viscous stresses and e=c v T+ 1 ( u 2 + v 2 + w 2) (5) 2 is the internal energ per mass unit. The governing equations are non-dimensionalied with the reference length L = (µ Re)/(ρ U ) and the freestream values of velocit, densit, temperature, viscosit and conductivit at the inflow. Note that the pressure is non-dimensionalied with ( ρ U ) 2. Air is treated as a non-reacting caloricall perfect gas with constant Prandtl number Pr=0.71 and constant specific-heat ratioκ=c p /c v = 1.4. The set of equations is closed b the equation of state p=(ρt)/(κ Ma 2 ). Sutherland s law is used to calculate the dnamic viscositµas a function of temperature [7]. In these equations, v represents the velocit vector, p labels the pressure, ρ indicates the densit, T is the temperature, and t denotes the time. The non-dimensional parameters are the Mach number Ma, the Prandtl number Pr and the Renolds number Re. The latter is defined as Re= ( ρ U L ) / ( µ ) and set to Re=10 5. I is the identit matri. The subscript refers to freestream values and the asterisk labels dimensional quantities. 2.2 Computational domain, boundar conditions and simulation setup In the present work, a laminar boundar-laer flow over a flat plat at an initial Mach number of Ma=2.67 with a favorable streamwise pressure gradient and an adiabatic-wall boundar condition is investigated. Figure 1-a shows the rectangular integration domain. Note that the weak shock wave emerging from the leading edge is not included. A time-accurate direct numerical simulation (DNS) is used to compute the flow quantities. A DNS solves the governing equations without turbulence modeling and allows for detection of an enhanced laminar-flow instabilit leading to self-ecited unsteadiness. The code uses splitted compact finite differences of 6 th -order in streamwise and wall-normal direction and a Fourier spectral ansat in spanwise direction, due to the assumed periodicit of the flow field. The classical eplicit 4 th -order Runge-Kutta procedure is applied for the integration in time. For more details of the code see [8]. a) b) (4) M U oo δ () 3 λ 0 0 c N Figure 1: a) Integration domain. b) Pressure distribution at the freestream boundar for cases REF and FPG. 2

3 M.Keller, M.J.Kloker. INFLUENCE OF A FAVORABLE STREAMWISE PRESSURE GRADIENT ON FILM COOLING A self-similarit solution of boundar-laer theor provides the flow variables which are prescribed at the inflow (= 0 ). At the outflow (= N ), a buffer region is used, where all instantaneous flow variables are smoothl ramped to the baseflow solution [8]. At the wall (= 0 ), the no-slip, no-penetration boundar conditions are imposed on the velocit components. The pressure is calculated according to p/ w = 0 and the densit is computed from the equation of state. The wall is adiabatic (for the code validation presented in Chapter 3.1 an isothermal wall is used). Note that thermal conduction within the wall is neglected in this stud. At the freestream boundar (= M ), a linear pressure distribution is prescribed (see Figure 1-b). All other flow variables at this boundar are computed such that the gradient along spatial characteristics is ero, ecept for the temperature which is computed from the equation of state. The simulations are performed in two steps. First, the baseflow solution is computed b prescribing a linear pressure distribution at the freestream. Figure 1-b illustrates the downstream development of the freestream pressure for the reference case with ero pressure gradient, in the following labeled as case REF, and for the case with streamwise favorable pressure gradient, indicated b case FPG. In order to smoothl connect the initial pressure distribution of case REF with the desired linear pressure distribution of case FPG a 5 th -order polnomial blending function is used. The pressure distribution was chosen such that the pressure at the outflow boundar corresponds to one half of the pressure at the inflow boundar, i.e. 50 percent pressure drop per 186 δ 0. In this equation,δ 0 = indicates the boundar-laer thicknessδ=u/u e = 0.99 at the inflow 0 = 1.8. As a result an accelerated boundar laer flow is obtained, where the Mach number increases form Ma=2.67 at the inflow ( 0 = 1.8) to Ma=3.10 at the outflow ( N = 7.5). In a second step, the blowing is turned on and the film cooling is simulated using the above mentioned baseflow simulations as initial condition. In order to reduce the computational costs the integration domain now etends from =5.48 to =6.86, with the row of holes placed at c = The blowing is realied b modeling the mass flu and temperature distribution over the holes using a polnomial of 5 th -order. In order to prescribe the same maimum and area-averaged mass flu as in an assumed Hagen-Poiseuille flow through a blowing pipe the modeled radius r mod has to be larger than the real radius of the hole r c (r mod = 7/4 r c ). For a detailed description of the modeling see e.g. [2]. Table 1 contains an overview of the simulation setup. These values are consistent with the eperiment and previous numerical investigations [2, 3, 4]. Table 1: Simulation setup parameters. Initial freestream Mach number Ma 2.67 [-] Freestream temperature T 1.00 [-] 564 [K] Recover temperature T rec 2.20 [-] [K] Freestream pressure p 0.10 [-] [bar] Prandtl number Pr 0.71 [-] Reference length L 1.00 [-] [mm] Hole radius r c [-] [mm] Modeled hole radius r mod [-] [mm] Distance from the leading edge c [-] [mm] Spanwise spacing s and periodicit lengthλ [-] 0.8 [mm] Core temperature T c,core [-] 293 [K] Maimum blowing rate (ρv) ma 0.105/0.21 [-] N N N [points] [-] 0 M [-] [-] t [-] 0 M [-] 0 N, [-] 0 N, w/o blowing [-] 3

4 ADVANCED NUMERICAL METHODS ID-162/ M.KELLER, M.J.KLOKER 3. Results 3.1 Code Validation For validation purposes a comparison with the numerical results of [9] and the eperimental findings of [6] is performed. According to the eperiment the wall is isothermal, T w = 293[K], and the streamwise pressure gradient is ero. Figure 2 shows the cooling effectiveness for blowing through infinite spanwise slits for various blowing ratios. The isothermal cooling effectiveness is defined asη is = 1 q c / q uc, where q c and q uc label the heat-flu distribution for the cases with and without cooling, respectivel. The comparison with the multi-grid finite-volume solver of [9] is carried out for a slit position of c = 5[m] and perpendicular blowing withα c = 90[ ]. For the comparison with the eperiment, the blowing is performed at an angle ofα c = 45[ ] and a slit position of c = 0.107[m]. The blowing ratio is given b (ρ v) av = (ρ v) av/(ρu) which corresponds to the integral mass flu, based on the velocit in blowing direction ( v). Both plots show an ecellent agreement. It can therefore be concluded that the code is well suited for the computation of film cooling in a supersonic flat-plate boundar laer. Note that the isothermal case with a coolant gas temperature equal to the (cool) wall temperature is a special case: It secures the laminarit of the boundar laer. 1.0 a) ρv av = b) ρv av = η is ρv av = 5 η is ρv av = [m] [m] Figure 2: Comparison of the isothermal cooling effectiveness. a) DNS ( ) vs. finite-volume solver [9] (smbols), Ma = 2.60, T = 488[K], p = [bar], c = 5[m],α c = 90[ ], ero pressure gradient. b) DNS ( ) vs. eperiment [6] (smbols), Ma = 2.61, T = 545[K], p = 0.15[bar], c = 0.107[m],α c = 45[ ], ero pressure gradient. 3.2 Streamwise Favorable Pressure Gradient In order to evaluate the baseflow results with a favorable streamwise pressure gradient obtained from DNS a comparison with parabolied Navier-Stokes simulations (PNS) is performed. Here, the stead Navier-Stokes equations are solved b marching in space neglecting all viscous terms with partial derivatives in downstream direction. For the PNS simulations the same parameter setup was chosen as for the numerical simulations. Figure 3 shows a comparison of the streamwise velocit, wall-normal velocit and densit distribution at = 5.6 and = 7.5. All three quantities are in good agreement. Figure 4 compares the streamwise and wall-normal velocit, densit and temperature distribution of the baseflow computations for the case with ero and streamwise pressure gradient at = 6.8. The graph also illustrates the downstream development of the skin friction coefficient, boundar-laer thickness, edge Mach number and recover factor. Note that the skin friction coefficient is computed according to c f =τ w /(0.5ρ e ()Ue 2 ()), using the edge values ofρ and u which are a function of for the case with favorable pressure gradient. Alternativel, a normaliation withρ and U can be emploed, leading to c f =τ w /(0.5ρ U 2 ) and resulting in a different behavior: Figure 4 shows that c f =τ w /(0.5ρ e ()Ue()) 2 is larger and slightl increases in downstream direction. Furthermore, it can be seen that the edge Mach number, the boundar-laer thickness, the skin friction coefficient, the u-velocit gradient at the wall and the streamwise velocit are increasing with decreasing freestream pressure. Unlike the Mach number, the streamwise velocit increases onl slightl (u FPG = 1.05 u REF at =6.8). Also, the wall-normal velocit component is increasing 4

5 M.Keller, M.J.Kloker. INFLUENCE OF A FAVORABLE STREAMWISE PRESSURE GRADIENT ON FILM COOLING 0.10 = 5.6 = 7.8 = 7.8 = 7.8 = u v = ρ Figure 3: Velocit and densit profiles obtained from DNS ( ) and PNS ( ) at =5.6 and =7.5 for the baseflow simulation with favorable streamwise pressure gradient. which is in contrast to the incompressible case. Note that u( ρ/ ) has the opposite sign of ρ( u/ ), its modulus is larger, and thus from the continuit equation ( v/ )>0 for all. In addition to that, it can be observed that the densit, the edge temperature and the adiabatic wall temperature are decreasing. The latter is due to increasing recover losses as the flow is accelerated. It holds T aw = T e (1 r)+rt 0 or r=(t aw T e )/(T 0 T e ), with r being the recover factor and T 0 = 1368[K] (see [7]). r slightl diminishes from to 0.827, i.e. the losses are slightl larger than with r=constant. The baseflow simulation provides the initial conditions for the simulation with film cooling. At first, the influence of an increased grid resolution is investigated. The results of the simulations for case FPG with a coarse and fine spanwise resolution ( f ine = 0.5 coarse ) and a blowing ratio ofρv ma = 0.21 are illustrated in Figure 5. It can be seen that all three quantities show a ver good agreement. The results are therefore considered to be grid independent. A higher resolution in streamwise and wall-normal direction leads to the same conclusion and is therefore not shown. In the following, all simulations are performed on the computational grid of the coarse simulation with points in -, - and -direction, respectivel. Figure 6 illustrates the wall-normal velocit distribution in a crosscut along the centerline of the hole ( = 0) for cases REF and FPG. The picture shows that the wall-normal blowing velocit (v) is increasing as the freestream 0.10 REF FPG c f u c f = τ w /(ρ 2 e U2 e ) c f = τ w /(ρ 2 U2 ) v 0.08 δ Ma e ρ T aw T T aw r Figure 4: Baseflow properties for the case with favorable streamwise pressure gradient and the reference case with ero pressure gradient. u-, v-, ρ- and T-profiles are taken at = 6.8. r 5

6 ADVANCED NUMERICAL METHODS ID-162 / M.KELLER, M.J.KLOKER 0.12 fine grid coarse grid = = = λ/2 = = λ/ u = λ/ ρ v Figure 5: Velocit and densit profiles at = 0 and = λ /2 for the coarse and fine -grid resolution at = c. Case FPG at a blowing rate of ρvma = 0.21, f ine = 0.5 coarse. pressure is decreasing. This can be eplained b the fact that, at a fied blowing rate (ρv), the wall-normal blowing velocit has to increase, since the densit (ρ) in the flow field and at the wall decreases (see Figure 4). As a result the wall-normal blowing momentum (ρv v) also has to increases. Note further that the compression-fan shocklet generated b the blowing gets stronger and bends towards the wall due to the larger Mach number for case FPG. The characteristic counter rotating vorte pair generated b the blowing is shown in Figure 7. The picture shows isosurfaces of constant λ2 [10] for cases REF and FPG at a blowing ratio of ρvma = The shading refers to the spanwise vorticit, where white indicates a clockwise rotation when looking downstream and black labels a counterclockwise rotation of the vortices. It can be seen that the higher wall-normal blowing momentum for case FPG results in a more pronounced counter rotating vorte pair. In addition, a horseshoe-vorte like structure at the leading edge of the hole can be observed which is not visible for the reference case. The temperature distribution and isolines of the streamwise velocit in a spanwise crosscut at = 6.35 are plotted in Figure 8 which also shows the isolines of λ2 = 10, corresponding to Figure 7. The picture displas the low-speed streak after the hole where the coolant fluid is transported awa from the wall, as well as both high-speed streaks between two holes where the hot gas from the boundar laer is transported towards the wall. Due to the streamwise pressure gradient the temperature distribution in the boundar laer has changed significantl, T is considerabl lower 0.15 b) FPG, ρvma = a) REF, ρvma = v c) REF, ρvma = 0.21 d) FPG, ρvma = Figure 6: Wall-normal velocit distribution along the centerline of the hole ( = 0) for cases REF and FPG at blowing rates of ρvma = and ρvma = 0.21 ( 0.12 < v < 0.12, v = 5). 6

7 M.Keller, M.J.Kloker. INFLUENCE OF A FAVORABLE STREAMWISE PRESSURE GRADIENT ON FILM COOLING a) REF, ρvma = b) FPG, ρvma = Figure 7: Vorte visualiation (bottom view) using isosurfaces of λ2 = 10 for cases REF and FPG at a blowing rate of ρvma = than for the reference case. Keep in mind that the integral mass flu of coolant gas is identical in both cases. The difference originates onl from the ongoing flow acceleration. In order to properl compare cases REF and FPG the cooling effectiveness is considered. This quantit relates the uncooled with the cooled scenario, i.e. the cases with and without blowing. The cooling effectiveness for an adiabatic wall is defined as ηad = (T aw,uc T aw,c )/(T aw,uc T c ). In this equation T aw,uc refers to the adiabatic wall temperature with the blowing turned off, T aw,c indicates the wall temperature of the cooled wall, and T c labels the temperature of the coolant gas. Both the spanwise averaged cooling effectiveness and adiabatic wall temperature for cases REF and FPG are shown in Figure 9. The quantities are plotted as a function of ξ = ( c ) / ρvc,av A which allows for a 2 better comparison of the cases with different blowing ratios (A = πrc ). A comparison of the different blowing rates reveals that the cooling effectiveness increases with an increasing blowing rate. An eception, however, is the the region right downstream of the holes, where the coolant gas is transported awa from the wall due to a stronger wall-normal blowing momentum. When comparing the cases with ero and favorable pressure gradient a similar behavior can be observed. Right downstream of the holes, the cooling effectiveness for case FPG is lower than for the corresponding reference case. Again, this can be eplained b the stronger wall-normal blowing momentum for case FPG, now due to the favorable streamwise pressure gradient. Despite the higher wall-normal blowing momentum for case FPG, the higher streamwise velocit close to the wall leads to the fact that the coolant gas is convected further downstream. This results in a higher cooling effectiveness for the cases with favorable streamwise pressure gradient, in particular for the case with ρvma = Here, the cooling effectiveness reaches values that are comparable to the reference case with ero pressure gradient and double blowing rate ρvma = However, the effect is less pronounced as the blowing ratio is increased which can be eplained b the overall higher blowing momentum. For case FPG with ρvma = 0.21 an improvement of the cooling effectiveness can be observed for the region of ξ & or & 6.59, respectivel. The downstream development of the spanwise averaged boundar-laer thickness for cases REF and FPG with and without blowing is presented in Figure 10. It can be observed that the boundar-laer thickness for the cases with 0.04 b) FPG, ρvma = 0.21 a) REF, ρvma = T Figure 8: Crosscut of the temperature flow field with isolines of u-velocit (0.0 < u < 1.0, u = ) and isolines of λ2 = 10 at = 6.35 for cases REF and FPG at a blowing rate of ρvma =

8 ADVANCED NUMERICAL METHODS ID-162/ M.KELLER, M.J.KLOKER η ad ξ = [- c ]/[(ρv) av A] T aw REF, ρv ma = FPG, ρv ma = REF, ρv ma = FPG, ρv ma = ξ = [- c ]/[(ρv) av A] Figure 9: Spanwise averaged cooling effectiveness and wall temperature for cases REF and FPG at blowing rates of ρv ma = andρv ma = blowing (ρv ma = 0.21) is considerabl larger than for the cases without blowing due to the coolant gas injection. However, the boundar-laer growth has vanished, since the injected gas leads to a significant cooling of the flat plate. Figure 11 illustrates the wall-normal velocit gradient (du/d 2 w+ dw/d 2 w) multiplied with the viscositµ. The quantit is directl proportional to the skin friction. We note that the variation of the viscositµis not significant. The figure demonstrates that a higher blowing rate results in a higher wall-normal velocit gradient (maimum and mean). This is due the more pronounced counter-rotating vorte pair generated b the stronger wall-normal blowing. Furthermore, it can be observed that the values for case REF are lower than for case FPG. The more pronounced counter-rotating vorte pair due to a stronger wall-normal blowing momentum can be hold responsible for this behavior. In addition to that, the streamwise flow acceleration results in increasing values of the velocit gradient in wall-normal direction. 4. Conclusions The present work is dedicated to the influence of a favorable streamwise pressure gradient on film cooling in a laminar Mach-2.67 boundar laer with an adiabatic wall-temperature condition. It is found that for a fied blowing ratio (ρv) the wall-normal blowing velocit increases, since the densit (ρ e,ρ w ) decreases with increasing downstream distance. Therefore, the wall-normal blowing momentum (ρv v) is stronger and a more pronounced counter-rotating vorte pair can be observed. However, the increased streamwise velocit close to the wall leads to a stronger downstream convection of the coolant gas and overcompensates the latter effect. As a result, a favorable streamwise pressure gradient leads to an increased cooling effectiveness. At the blowing rates considered no global instabilit could be observed. Future work is going to focus on the influence of a transitional and turbulent boundar laer state. Furthermore, an evaluation of the modeled blowing approach is going to be performed b a comparison with simulated pipes δ 0.08 REF, w/o blowing FPG, w/o blowing REF, ρv ma = 0.21 FPG, ρv ma = c Figure 10: Spanwise averaged boundar-laer thickness for cases REF and FPG with and without blowing. 8

9 M.Keller, M.J.Kloker. INFLUENCE OF A FAVORABLE STREAMWISE PRESSURE GRADIENT ON FILM COOLING a) µ 2 2 du/dw + dw/dw: REF, ρvma = FPG, ρvma = b) REF, ρvma = FPG, ρvma = Figure 11: Wall-normal velocit gradient multiplied with the viscosit for cases REF and FPG at blowing rates of ρvma = and ρvma = Acknowledgments The financial support of the German Research Foundation (Deutsche Forschungsgemeinschaft) in the framework of the Special Research Center SFB/TRR 40 and the computational resources, kindl provided b the Federal High Performance Computing Center Stuttgart (HLRS), are gratefull acknowledged. Also, special thanks go to T. Goten and M. Hombsch (both RWTH Aachen) for their numerical and eperimental reference data. References [1] Linn, J. and Kloker, M.J. (2008). Numerical Investigations of Film Cooling and its Influence on the Hpersonic Boundar-Laer Flow. In: Gülhan, A. (Ed.), RESPACE - Ke Technologies for Reusable Space Sstems, NNFM, 98, , Springer. [2] Linn, J. and Kloker, M.J. (2011). Effects of Wall-Temperature Conditions on Effusion Cooling in a Mach-2.67 Boundar Laer. AIAA-J., 49/2, , DOI: /1.J [3] Linn, J., Keller, M. and Kloker, M.J. (2010). Effects of Inclined Blowing on Effusion Cooling in a Mach-2.67 Boundar Laer. Sonderforschungsbereich/Transregio 40 - Annual Report 2010, [4] Heufer, K.A. and Olivier, H. (2008). Eperimental Stud of Active Cooling in 8 Laminar Hpersonic Flows. In: Gülhan, A. (Ed.), RESPACE - Ke Technologies for Reusable Space Sstems, NNFM, 98, , Springer. [5] Heufer, K.A. and Olivier, H. (2008). Eperimental and Numerical Stud of Cooling Gas Injection in Laminar Supersonic Flow. AIAA-J., 46/11, , DOI: / [6] Hombsch, M. and Olivier, H. (2010). Flow condition and cooling gas variation for film cooling studies in hpersonic flow. Sonderforschungsbereich/Transregio 40 - Annual Report 2010, [7] White, F.M. (1991). Viscous Fluid Flow. McGraw-Hill. 9

10 ADVANCED NUMERICAL METHODS ID-162/ M.KELLER, M.J.KLOKER [8] Babucke, A., Linn, J., Kloker, M.J. and Rist, U. (2003). Direct Numerical Simulation of Shear Flow Phenomena on Parallel Vector Computers. In: Resch, M. et al. (Eds.), High Performance Computing on Vector Sstems. High Performance Computing Center, Stuttgart (HLRS), Springer. [9] Dahmen, W., Goten, T.and Müller, S. (2009). Numerical Simulation of Cooling Gas Injection Using Adaptive Multiscale Techniques. Sonderforschungsbereich/Transregio 40 - Annual Report 2009, [10] Joeng, J.and Hussian, F. (1995). On the Identification of a Vorte. J. Fluid Mech., 285,