DNS CODE VALIDATION FOR COMPRESSIBLE SHEAR FLOWS USING LST

Transcription

1 Copright c 2016 ABCM DNS CODE VALIDATION FOR COMPRESSIBLE SHEAR FLOWS USING LST Jônatas Ferreira Lacerda, jonatasflacerda@hotmail.com Leandro Franco de Souza, lefraso@icmc.usp.br Josuel Kruppa Rogenski, josuelkr@gmail.com Institute of Mathematical and Computer Sciences, Universit of São Paulo. São Carlos-SP, Brazil. Márcio Teieira de Mendonça, marciomtm@iae.cta.br Instituto de Aeronáutica e Espaço, Divisão de Propulsão, São José dos Campos, SP, Brazil Abstract. In order to simulate compressible shear flow stabilit and aeroacoustic problems, a numerical code must be able to capture how a baseflow behaves when submitted to known small disturbances. If the disturbances are amplified the flow is unstable. The Linear Stabilit Theor (LST) provides a framework to obtain information about the growth rate in relation to the ecitation frequenc for a given base flow. The correspondent eigenfunction when used for disturbance introduction on direct numerical simulations (DNS) should capture the same growth rate. In the present work, DNS simulations of a two-dimensional compressible miing laer and a two-dimensional compressible plane jet are performed. Disturbances are introduced at the domain inflow and the spatial growth rate obtained with a DNS code are compared with growth rates obtained from a LST analsis for each baseflow in order to verif and validate the DNS code. The good comparisons between the DNS simulations and LST results indicate that the code is able to simulate compressible flow problems and it is possible to use it to perform direct numerical simulation of instabilit and aeroacoustics problems. Kewords: compressible flow, direct numerical simulations, linear stabilit theor, code validation 1. INTRODUCTION There is an increasing concern about health problems related to noise in man engineering areas. Some noise sources are due to turbulence in a flow field and aeroacoustics is the area responsible for the investigation of noise generation and propagation b flow (aerodnamic noise). The problem of flow generated noise have some specific characteristics [9]: multiple frequencies, multiple scales and propagation to long distances with ver low dissipation. To deal with these characteristics a numerical simulation must [9]: have a transient numerical scheme, have ver low numerical dissipation and dispersion to preserve the propagation of acoustic waves and have non-reflecting boundar conditions when the phsical domain is truncated to form the computational domain. Along the ears man studies has been conducted in the area of Computational Aeroacoustics (CAA). Two main approaches can be identified: The first one rel on acoustic analogies ([6], [7], [2],[3]) where noise sources are obtained from CFD results and its propagation are calculated solving an inhomogeneous wave equation for pressure subjected to these sources. The second approach uses a high fidelit simulation capable of computing directl the acoustic waves together with the fluid flow, which is called direct numerical simulation (DNS) of aeroacoustics. In order to be able to perform DNS of aeroacoustic problems in the future, a numerical code was developed within our research group. It uses parallel processing with domain decomposition to perform the calculations in a feasible time. High order temporal and spatial discretization schemes were adopted to minimize the dispersion and dissipation phenomena on the simulation of the flow field and of the acoustics waves. A series of strategies such as filtering and mesh stretching as well as characteristic boundar conditions were implemented in order to obtain a proper solution of the aeroacoustics problem. However, before using this code for aeroacoustic predictions, it is necessar to verif and validate it and see if the code is able to capture the flow behaviour when submitted to known small disturbances. The Linear Stabilit Theor (LST) provides a framework to obtain information about the amplification rate in relation to the ecitation frequenc and the corresponding eigenfunctions. The results from LST can be used for comparisons with DNS simulations. In the present work results are presented for DNS simulations of a two-dimensional compressible miing laer and two-dimensional compressible plane jet. Disturbances were introduced at the domain inflow and the code verification was done b comparing the spatial amplification rate obtained b the DNS code with amplification rates obtained from a LST analsis of each base flow. The good comparison obtained indicated that the code can simulate compressible flow problems and it is possible to use it to perform direct numerical simulation of aeroacoustics problems.

2 Copright c 2016 ABCM 2. FORMULATION AND NUMERICAL SCHEME The code was developed to solve the two-dimensional unstead compressible Navier-Stokes equations together with the continuit and energ equations in the dimensionless form ([5]). A perfect gas relation for pressure is assumed to close the problem. In the conservative form, the unknowns are given b the densit ρ, momentum densities ρu, ρv and total energ E. Time integration is performed using a 4th-order 4-steps Runge-Kutta scheme. The spatial discretization in the streamwise and normal directions is done b a 6th-order compact finite difference method and the generated tridiagonal equation sstems are solved using the Thomas algorithm (TDMA). An non-uniform, grid is used, as shown in Fig. 3. An interest region with uniform grid is delimited. Then, a grid stretching of approimatel 1% is applied in both directions that, together with spatial low-pass filtering, creates a damping zone. Disturbances become increasingl badl resolved and dissipated as the propagate through the damping zone before the reach the outflow boundaries. The code was parallelized through domain decomposition in both directions using shared-memor parallelization and MPI libraries are used for inter nodal communications. A grid transformation in the (, ) plane is used to map the phsical grid in a equidistant (ξ, η) computational grid, and metrics are necessar for the derivative calculations. The first derivatives are given b: = ( 1 ) ξ, ξ = ( 1 ) η. η Second derivatives are: 2 2 = 2 2 = ( 1 ξ 1 ( η ) 2 2 ξ 2 ) 2 2 η 2 where the metrics are given b: ( 2 ξ 2 ξ 2 η 2 ( η ) 3 ξ = 2 ξ 2 ) 3 η = 2 η 2 ( ) 2 ξ 2 ξ ξ 2, (3) ( ) 2 η 2 η η 2, (4) (1) (2) 2 ξ 2 = 2 ξ 2 2 η 2 = 2 η 2 ( ) 3 ξ, ( ) 3 η. Special boundar conditions are used to avoid disturbance reflection from boundaries. For the inflow, characteristics are used and for the free stream and outflow boundaries null derivative of the properties are used. Together with grid stretching and low-pass filtering, the dissipate the disturbances before the reach the boundaries. Inviscid linear stabilit theor analsis was used to obtain the frequenc with maimum spatial growth rate and the correspondent eigenfunctions are introduced as disturbances at the simulation inflow boundar. 3. RESULTS 3.1 Compressible two-dimensional miing laer A compressible miing laer consists of two streams with different velocities flowing parallel to one another such that the resulting velocit profile assumes an S shape. Due to the inflectional point on the velocit profile, the flow is unstable and transition to turbulence due to the generation and growth of vortical structures. For the present verification, the flow configuration has been closel matched to the case investigated b [1]. The Mach numbers of the upper and the lower streams are Ma 1 = 0.50 and Ma 2 = 0.25, respectivel. The Renolds number is 500 and is based on the vorticit thickness at the inflow. The inflow velocit, densit and temperature profiles are showed in Fig. 1 and are obtained b solving the stead compressible two-dimensional boundar-laer equations. The eigenfunctions used to introduce the disturbances at the DNS inflow boundar were obtained from a spatial viscous linear stabilit analsis, where the fundamental frequenc is ω = The corresponding eigenfunction amplitude and (5) (6)

3 Proceedings of EPTT 2016 Copright c 2016 ABCM u r T 03 Figure 1. Inflow profile for velocit u, densit and temperature for the 2D miing laer. phase are- shown in Fig. 2. This and- three subharmonics were used Perfil - to disturb the flow. The entrada - maimum amplitude of the fundamental - frequenc is 02 while- for the other subharmonics - the maimum amplitude is - 01, all normalized b the correspondent 0 maimum u0.75 velocit 0 amplitude. 0.1 Furthermore, 0.2 Camada ade phase 00 mistura shift20 of 2D θ = was introduced for 15 the first subharmonic, θ = ûu for the second and θ = ˆv v for the third. r ˆr T ˆ T u u v v - - (a) r Amplitude e fase perturbação Camada de mistura 2D r T T u û v ˆv ˆ ˆT Figure 2. (a) amplitude and (b) Amplitude phase distribution e fase perturbação for the 2D miing laer. Camada de mistura 2D A representation of the computational mesh, showing the domain regions and boundar conditions, is presented in Fig. 3. A computational mesh with points in the and directions was used. In the longitudinal direction the mesh has constant increment = up to = 300, forming the region of interest and downstream the mesh is stretched- with a rate of approimatel %, forming a damping - zone. In the normal direction - the mesh is stretched from the center - to the boundaries, with the- lowest increment of = and the highest increment - of = 6. As inflow - boundar conditions, - the u velocit, temperature- and densit profiles presented - in Fig. 1 were used. The disturbances- were -1.5 introduced 1.5 at each step - of -1.5 the Runge-Kutta. 1.5 Furthermore, characteristic 1.5 boundar conditions 1.5 are used at the inflow to avoid reflections of outgoing waves back to the computational domain. To minimize reflections caused b oblique acoustic waves, a damping zone is applied at the upper and lower boundar, forcing the flow variables to a stead state solution. To avoid large structures flowing through the outflow boundar, a combination of grid stretching (b) T

4 Copright c 2016 ABCM Free stream Damping zone Grid stretching Spatial filtering in and Inflow Interest region Outflow Free stream Figure 3. Computational domain detaching the interest region, the damping zone and the boundaries for 2D miing laer. and spatial low-pass filtering are applied in the damping zone. Thus, disturbances become increasingl badl resolved as the propagate through this region and b appling a spatial filter, the perturbations are substantiall dissipated before the reach the outflow boundar. The chosen time-step was t = 2π ω n steps, where n steps is the number of points per wavelength of the fundamental frequenc. In this case, n steps = 752, corresponding to a time-step t = 133. Sevent si periods of the fundamental frequenc were simulated and the last eight were chosen for post-processing analsis. The observation of the maimum amplitude of the normal velocit v along the direction for each position can show how the disturbances are growing in the domain, as shown in Fig. 4. In the initial part of the domain the amplitudes grow eponentiall until non-linear effects are observed, causing a saturation of the maimum amplitude of v. The spatial growth rate α i, calculated b Eq. 7, can be compared with results from linear stabilit theor, as shown in Fig. 5. Table 1 presents the difference between DNS and LST results, where there is an initial portion of the domain with an elevated error due the receptivit of the flow to the disturbances near the inflow. Further downstream (after 45) the differences decrease and the maimum difference was 4% for mode (1,0), 9% for mode (1/2,0) and 15% for mode (1/4,0). Although α i is a ver sensitive value, mainl for the subharmonics, the mean values of the DNS corresponded well to those predicted b linear stabilit theor. Thus, the good agreement between simulation and theor serves as a verification of the implemented DNS computational code for small disturbances. α i = [ln ( v ma )]. Besides the spatial amplification rate comparison, the transversal vorticit Ω z at the end of the simulation were compared with the results presented b [1] as shown in Fig. 6. It is possible to see the formation of the first Kelvin-Helmholtz vortices which pair downstream forming new structures. A good match with [1] was obtained. (7)

5 ai a Proceedings of EPTT 2016 Copright c 2016 ABCM 0E+00 0E-01 0E-02 0E-03 0E-04 0E DNS - (1,0) DNS - (1/2,0) DNS - (1/4,0) Babucke - (1,0) Babucke - (1/2,0) Babucke - (1/4,0) Figure 4. Maimum amplitude of the normal velocit v along the direction for each position comparison with results from [1] DNS - (1,0) DNS - (1/2,0) DNS - (1/4,0) LST - (1,0) LST - (1/2,0) LST - (1/4,0) Figure 5. Spatial amplification rate comparison between LST and DNS for the 2D miing laer. Table 1. Spatial amplification rate comparison between LST and DNS for the 2D miing laer. (1,0) (1/2,0) (1/4,0) LST DNS % LST DNS % LST DNS % % % % % % % % % % % % % % % % % % % % % % % % %

6 Copright c 2016 ABCM (a) (b) Figure 6. Transversal vorticit at final time-step. Contour levels from up to 2 with an increment of 4. (a) results from [1] and (b) DNS code. 3.2 Compressible two-dimensional plane jet Plane jets are a tpical shear flow present in several applications such as in combustion and propulsion. The interest in aerodnamic noise generated b jets is growing fast in the aeronautical industr. In this flow, a central core with high velocit mies with a parallel stream with lower velocit. In the interface between both flow streams there is high gradients of the flow properties, making the flow unstable to disturbances. In the present investigation a two-dimensional plane jet was studied. The inflow velocit profile is given b: u () = Ma j Ma Ma j + Ma tanh [2 ( + θ)] + 1, (8) where θ = 1.98 and the velocit profile is smmetric with respect to = 0 as shown in Fig. 7. The ratio between the jet and the far field co-flow was kept equal to U j /U = 1.67 for all cases. The jet is perfectl epanded and isothermal such that the temperature, densit and pressure profiles are kept constant. The Prandtl number is Pr = Since the linear stabilit analsis is inviscid, the DNS simulations are done with a high Renolds number (Re = 10000) such that the inertia effects are much higher than the viscous effects. A computational Cartesian grid with nodes in and directions was used in all simulations. In the longitudinal direction the grid is uniform with = 0.25 up to 250 and than stretched at a % rate. In the normal direction, there is also a uniform grid with = up to ±5.20 and then stretched at a % rate. The uniform spacing zone corresponds to the interest zone, while the stretched grid region corresponds to the damping zone. Domain decomposition was adopted to parallel the computation. For boundar conditions, at inflow the u velocit component is given b Eq. 8. As in the two-dimensional miing laer case, the disturbances were introduced at each step of the Runge-Kutta scheme and the necessar numerical cares adopted to avoid numerical contamination of the solution was also used. The chosen time step was t = 2π ω n steps, where n steps is the number of points per wavelength of the fundamental frequenc. The number of time steps used b fundamental frequenc was n steps = 752. In each case 76 periods were simulated for the fundamental frequenc, where the last 8 were chosen to post-processing analsis. Compressible jet stabilit results are know to have two possible modes, a varicose mode and a sinuous mode. An inviscid LST analsis was performed considering onl the varicose mode. A classic normal modes analsis using the Gropengiesser [4] formulation was performed, where a new variable ξ is defined combining both p and v disturbance eigenfunctions as ξ = iαp γma 2 v g = α2 + β 2 2 (αu ω)2 ρα 2 Ma α 2, (9) (10)

7 alfa i Proceedings of EPTT 2016 Copright c 2016 ABCM u Figure 7. u-velocit profile for the 2D plane jet. resulting in the following form of the Raleigh stabilit equation ([8]). ξ = ρα(αu ω) Perfis de entrada Biringen 2D αξ αu ω (gξ U ). (11) Where α and ω are the streamwise wave number and frequenc. The simulated cases and their correspondent maimum spatial amplification rate and frequenc are presented in Tab. 2. The corresponding jet, free-stream and convective Mach numbers (Ma j, Ma, and Ma c ) are shown on the table. Ma c = Ma j Ma. 2 (12) Table 2. Mach numbers and maimum spatial amplification rates. Case Ma j Ma Ma c ω r α r α i The variation of the spatial amplification rate with Ma j is shown in Fig. 8. It is possible to observe that increasing the Mach number there is a decrease in the maimum value of the amplification rate, and the flow becomes more stable i omega r Maj=0,25 Maj=1,05 Maj=1,50 Maj=2,00 Maj=2,50 Maj=3,00 Figure 8. Spatial amplification rates for the 2D plane jet. The decrease of the amplification rate and consequent miing of the flow is a known effect. This effect is of great importance in combustion flows and can also help on noise control. taas biringen 2D varicoso

8 Copright c 2016 ABCM The maimum amplitude of the normal velocit v along the direction as a function of the streamwise position ( v ma ) are presented in Fig. 12. It is possible to identif an initial receptivit region of the disturbances near the inflow. Downstream occurs an eponential increase in the amplitude corresponding to linear growth effects up to a maimum level were non-linear effects become significant. As the flow becomes more stable with increasing Mach number, the position where ( v ma ) reaches a maimum is delaed to a position further downstream with the increase of the Mach number. A comparison with spatial amplification rates obtained b the LST analsis is also presented in Figs. 12. Ecept for the receptivit region close to the inflow, there is a good comparison between LST and DNS results. Again, it was possible to sa that the good agreement between simulation and theor serves as a verification of the implemented DNS computational code for small disturbances. 0E+00 Maj=0.25 0E+00 Maj=5 0E-01 0E-01 0E-02 0E-02 0E-03 0E-03 0E-04 0E-04 0E-05 0E-05 0E-06 0E-06 0E+00 Maj=1.50 0E+00 Maj=0 0E-01 0E-01 0E-02 0E-02 0E-03 0E-03 0E-04 0E-04 0E-05 0E-05 0E-06 0E-06 0E+00 Maj=2.50 0E+00 Maj=0 0E-01 0E-01 0E-02 0E-02 0E-03 0E-03 0E-04 0E-04 0E-05 0E-05 0E-06 0E-06 DNS LST Figure 9. Maimum amplitude of the normal velocit v along the direction for each position (DNS result) and comparison with LST spatial amplification rate. varicoso

9 Copright c 2016 ABCM 4. SUMMARY Two compressible shear flows has been simulated using a two-dimensional DNS code, a two-dimensional compressible miing-laer and a two-dimensional compressible plane jet. The results have been compared with linear stabilit theor and ver good agreement have been found, showing that the DNS code can simulate wave propagation phenomena with the necessar accurac regarding dispersion and dissipation effects. 5. ACKNOWLEDGEMENTS For their support, Tecumseh do Brasil Ltda. and National Council for Scientific and Technological Development (CNPq). 6. REFERENCES Babucke, A. and Kloker M. and Rist U., 2008, DNS of a Plane Miing Laer for the Investigation of Sound Generation Mechanisms, Computers and Fluids, Vol.37, n.4, pp Curle, N., 1955, The Influence of Solid Boundaries upon Aerodnamic Sound, Proceedings of the Roal Societ of London. Series A. Mathematical and Phsical Sciences, Vol.231, n.1187, pp Ffowcs-Williams, J.E. and Hawkings, D.L., 1969, Sound Generation b Turbulence and Surfaces in Arbitrar Motion, Philosophical Transactions of the Roal Societ of London. Series A. Mathematical and Phsical Sciences, Vol.264, n.1151, pp Gropengiesser, H., 1970, Stud on the stabilit of boundar laers in compressible fluids, Tech. Rep. NASA TT F-12, National Aeronautics and Space Administration NASA. Lacerda, J.F., 2016 Aeroacústica Computacional através de Simulação Numérica Direta de Escoamentos Livres Cisalhantes Compressíveis, PhD Thesis, ICMC - USP, São Carlos-SP, Brazil Lighthill, M.J., 1952, On Sound Generated Aerodnamicall. I. General Theor, Proceedings of the Roal Societ of London. Series A. Mathematical and Phsical Sciences, Vol.211, n.1107, pp Lighthill, M.J., 1954, On Sound Generated Aerodnamicall. II. Turbulence as a Source of Sound, Proceedings of the Roal Societ of London. Series A. Mathematical and Phsical Sciences, Vol.222, n.1148, pp Salemi, L. C., and Mendonca, M. T., 2008, Spatial and Temporal Linear Stabilit Analsis of Binar Compressible Shear Laer, AIAA 38th Fluid Dnamics Conference, AIAA paper , Seattle, USA. Tam, C.K.W., 2004, Computational Aeroacoustics: An Overview of Computational Challenges and Applications, Int. Journal of Computational Fluid Dnamics, Vol.18, n.6, pp