Governing equations for a new compressible Navier-Stokes solver in general cylindrical coordinates

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1 UNIVERSITY OF SOUTHAMPTON SCHOOL OF ENGINEERING SCIENCES AERODYNAMICS & FLIGHT MECHANICS GROUP Governing equations for a new compressible Navier-Stokes solver in general cylindrical coordinates by Richard D. Sandberg Report No. AFM-7/7 November 27 COPYRIGHT NOTICE - All right reserved. No parts of this publication may be reproduced stored in a retrieval system or transmitted in any form or by any means electronic mechanical photocopying recording or otherwise without the permission of the Head of the School of Engineering Sciences University of Southampton Southampton SO17 1BJ U.K. c School of Engineering Sciences University of Southampton

2 The aim of this report is to derive the governing equations for a new compressible Navier- Stokes solver in general cylindrical coordinates i.e. the streamwise and radial directions are mapped to general coordinates. A literature review revealed that simulations of complex cylindrical geometries have mostly been conducted using general curvilinear coordinates e.g. DeBonis & Scott 22. The method presented in this report is chosen over three-dimensional general curvilinear coordinates as it requires a smaller number of terms to be computed. Furthermore only two-dimensional metric terms need to be stored reducing the required allocated memory and therefore resulting in a more efficient code. 1 Governing equations in cylindrical coordinates The compressible Navier-Stokes equations consist of conservation of mass momentum and total energy. The flow is assumed to be an ideal gas with constant specific heat coefficients. All quantities are made dimensionless using the flow quantities at a reference location in the flow; here the free-stream/inflow location is used. The radius of the body is chosen as the reference length. The non-dimensionalization results in the following dimensionless parameters: Re = ρ u r µ M = u a P r = µ c p κ With z r θ denoting the streamwise radial and azimuthal directions respectively and u v w denoting the velocity components in z r θ directions respectively the non-dimensional compressible Navier Stokes equations in cylindrical coordinates are A = C = U t + A z + B r + 1 r ρu ρuu + p τ zz ρuw τ θz ρuh + q z uτ zz vτ rz wτ θz ρw ρuw τ θz w τ θr ρww + p τ θθ ρwh + q θ uτ θz vτ θr wτ θθ U = ρ ρu ρw ρe. C θ + 1 r D = 1 B = D = v + p τ rr w τ θr H + q r uτ rz vτ rr wτ θr v ρww τ rr + τ θθ 2w 2τ θr H + q r uτ rz vτ rr wτ θr 2 where the total energy is defined as E = T/γγ 1M 2 + 1/2u i u i with γ = 1.4 and the total enthalpy is H = E + p/ρ. 1

3 The molecular stress tensor components are τ zz = 2µ 2 u z v r 1 w r θ + v 3 τ rr = 2µ u z + 2v r 1 w r θ + v 4 τ θθ = 2µ u z v w r + 21 r θ + v 5 τ rz = µ u Re r + v 6 z τ θz = µ w Re z + 1 u 7 r θ τ θr = µ 1 v Re r θ w + w. 8 r The heat-flux vector components are q z = q r = q θ = µ T P rγ 1M 2 Re z 9 µ T P rγ 1M 2 Re r 1 µ 1 T P rγ 1M 2 Re r θ 11 where the Prandtl number is assumed to be constant at P r =.72. The molecular viscosity µ is computed using Sutherland s law c.f. White 1991 setting the ratio of the Sutherland constant over freestream temperature to To close the system of equations the pressure is obtained from the non-dimensional equation of state p = ρt /γm 2. 2 Transformation to general cylindrical coordinates In order to allow for complex geometries the r z plane is mapped to general coordinates ξ η. Hence streamwise and radial derivatives need to be expressed in terms of the new variables. By using the chain rule the following expressions can be derived z r = 1 r J η = 1 z J η ξ r ξ ξ + z ξ η η = rη ξ r ξ η 12 = zξ η z η ξ 13 where J is the determinant of the coordinate transformation. For conciseness the metric terms are abbreviated as e.g. r η = r so that J = z η ξr η z η r ξ and the asterisk denotes a metric term already divided by J. 2

4 Following Anderson 1995 the r and z derivatives of equation 1 can be transformed as follows provided that the grid transformation does not vary in time U t = 1 J ξ Ar η Bz η + η Ar ξ + Bz ξ 1 C r θ 1 r D 14 Unfortunately it is not straightforward to plug in the vectors A and B into 14 as they contain terms that are pre-multiplied by 1/r. As r = rξ η when taking ξ and η derivatives of A and B the chain rule needs to be applied. To illustrate this procedure the derivation of the radial momentum equation is chosen as example. Here we only focus on the derivatives in the ξ and η directions r η v + p τ rr z η ξ r ξ v + p τ rr z ξ. 15 η The only term containing 1/r is the stress tensor component τ rr. Therefore the equation is separated into r η v + p 2µ u ξ z + 2v z η 1 2µ w r ξ r θ + v z η r ξ v + p 2µ u η z + 2v z ξ + 1 2µ w r η r θ + v z ξ. 16 Applying the chain rule to the last term of each row results in 1 2µ w ξ r θ + v z η = 1 2µ w r ξ θ + v z η 1 2µ w η r θ + v z ξ = 1 2µ w r η θ + v z ξ 1 r 2 r ξz η 2µ 1 r 2 r ηz ξ 2µ w θ + v w θ + v Substituting these two expressions into 16 yields r η v + p 2µ u ξ z + 2v z η r r ξ v + p 2µ u η z + 2v z ξ r 1 2µ w r ξ θ + v z η + 1 2µ w r η θ + v z ξ + 1 2µ w r 2 θ + v r ξ z η r η z ξ. 17 }{{} J The last term of 17 occurs because of the partial derivative with respect to r of τ rr. In the case of streamwise derivatives z we get r ξr η r η r ξ = hence 1/r 2 terms cancel. The resulting governing equations are now summarized. For brevity the inner derivatives are not written in terms of ξ η however the streamwise and radial derivatives need to be computed with equations 12 and 13. 3

5 2.1 Continuity equation ρ t = 1 J ξ ρur η z η + η ρur ξ + z ξ 1 ρw r θ r Streamwise momentum equation ρu t = 1 { ρuu + p 2µ 2 u J ξ z v r ρuu + p 2µ 2 u η z v r + 1 2µ w r ξ θ + v r η 1 2µ r η 1 ρuw τ θz 1 r θ r r η z η r ξ z ξ } w θ + v r ξ Radial momentum equation t = 1 { r η J ξ r ξ η 1 2µ w r ξ θ + v z η + 1 2µ w r 2 θ + v 1 r θ w τ θr 1 r v + p 2µ u z + 2v z η r v + p 2µ u z + 2v z ξ r + 1 } 2µ w r η θ + v z ξ v ww 2µ Re v r 1 w r θ + v 2 4

6 2.4 Azimuthal momentum equation ρw t = 1 J { ξ η ρuw µ Re w r η z w r ξ w µ Re ρuw µ w µ Re z Re r 1 µ u v r ξ Re θ r η θ w z η + 1 r η 1 µ v r 2 Re θ w 1 ρww + p τ θθ 1 2w 2τ θr r θ r w z η r w z ξ 21 } µ u v Re θ r ξ θ w z ξ 2.5 Energy equation ρe t { ρuh + q z u 2µ ξ 2 u z v r 2 v r u z w z = 1 vτ rz w µ r η 22 J Re H + q r uτ rz v 2µ w µ w z η Re r ρuh + q z u 2µ 2 u η z v vτ rz w µ w r ξ r Re z H + q r uτ rz v 2µ 2 v r u w µ w z ξ z Re r + 1 µ u 2 w r ξ Re 3 θ + v w u r η θ µ 2 w v Re 3 v θ + v w θ w z η 1 µ u 2 w r η Re 3 θ + v w u r ξ θ µ } 2 w v Re 3 v θ + v w θ w z ξ + 1 µ 2 w v r 2 Re 3 v θ + v w θ w 1 ρwh + q θ uτ θz vτ θr wτ θθ 1 H + q r uτ rz vτ rr wτ θr r θ r 5

7 3 Vector form of general cylindrical coordinates The equations in general cylindrical coordinates can be written as where  r = U t = 1 J { µ Re u r  = ˆB = 2µ ξ ξ Ârη ˆBz η + Âr ξ + η ˆBz ξ Âr r η ˆB r z η + η  r r ξ + ˆB r z ξ } + 1 r C θ 1 r D 1 r ˆB 2 rr 23 ρuh + q z u 2µ ρu ρuu + p 2µ 2 u v z r ρuw µ Re 2 u v z H + q r uτ rz v 2µ w + θ v µ Re w u θ θ + v w u θ ˆB rr = w z r vτrz w µ Re ρuv τ rz v + p 2µ u + 2 v z r w µ w Re r u + 2 v z r ˆBr = µ Re w v µ Re 2µ w + θ v µ v Re θ w w z w µ w Re r 2µ µ Re θ w 2 3 v w θ + v w + θ v v θ w 2 3 v w θ + v w v. θ w In order to make the new code most efficient a good compromise between the number of arithmetic operations and the number of three dimensional arrays stored needs to be found. The emphasis here lies on reducing the number of stored arrays to a minimum while minimizing the number of repeated operations. One choice is to simply store all nine velocity derivatives. However for this option the number of operations is high as the stress tensor components constantly need to be reassembled. The most efficient choice of 6

8 arrays to store appears to be the following set 1 τ zz = 2µ 2 u z v 2 τ rz = µ u r Re r + v z 4 τ rr = 2µ 2 v r u 5 τ θr = µ w z Re r 7 τ 2 = µ u 8 τ 3 = µ v Re θ Re θ w. 3 τ θz = µ w Re z 6 τ 1 = 2µ w θ + v Using these eight arrays in addition to the primitive variables and the heat-flux vector  ˆB Âr ˆB r and ˆB rr can be assembled. In order to evaluate C and D the following relations can be used τ θz = τ θz + 1 r τ 2 τ θr = τ θr + 1 r τ 3 τ rr = τ rr 1 r τ 1 τ θθ = τ zz τ rr + 2 r τ Finally all vectors required for 23 can be written as ρu ρuu + p τ zz  = ˆB = v + p τ rr ρuw τ θz w τ θr ρuh + q z u τ zz vτ rz w τ θz H + q r uτ rz v τ rr w τ θr τ 1  r = ˆBr = ˆBrr = τ 2 uτ 1 wτ 2 C = References τ 1 τ 3 vτ 1 wτ 3 τ 1 τ 3 wτ 3 vτ 1 ρw ρuw τ θz + 1τ r 2 w τ θr + 1τ r 3 ρww + p τ zz τ rr + 2τ r 1 ρwh + q θ u τ θz + 1τ r 2 v τ θr + 1τ r 3 w τ zz τ rr + 2τ r 1 D = v ρww τ rr 1 r τ 1 + τ zz τ rr + 2 r τ 1 2w 2 τ θr + 1 r τ 3 H + q r uτ rz v τ rr 1 r τ 1 w τ θr + 1 r τ 3. Anderson J Computational Fluid Dynamics The Basics with Applications. McGraw-Hill. DeBonis J. & Scott J. N. 22 Large-eddy simulation of a turbulent compressible round jet. AIAA J White F. M Viscous Fluid Flow. McGraw Hill. 7