The aim of this report is to derive generalized coordinates for the specific case of mapping only the streamwise and radial coordinate of a cylindrical coordinate system, while leaving the azimuthal coordinate unchanged. The characteristic equations and the required matrices for the transformation from conservative to characteristic form are presented for this specific case. All equations and procedures are based on previous work on generalized characteristic boundary conditions (Kim &amp; Lee, 2000) and characteristic interface conditions (Kim &amp; Lee, 2003).<br/

Sandberg, Richard D.

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UNIVERSITY OF SOUTHAMPTONSCHOOL OF ENGINEERING SCIENCESAERODYNAMICS & FLIGHT MECHANICS GROUPGeneralized cylindrical coordinates forcharacteristic boundary conditions andcharacteristic interface conditionsbyRichard D. SandbergReport No. AFM-09/02November 2009COPYRIGHT NOTICE -All right reserved. No parts of this publication may be reproduced, stored in a retrievalsystem, 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 EngineeringSciences, University of Southampton, Southampton SO17 1BJ, U.K.c ° School of Engineering Sciences, University of SouthamptonThe aim of this report is to derive generalized coordinates for the speci¯c case of mappingonly the streamwise and radial coordinate of a cylindrical coordinate system, while leav-ing the azimuthal coordinate unchanged. The characteristic equations and the requiredmatrices for the transformation from conservative to characteristic form are presented forthis speci¯c case. All equations and procedures are based on previous work on general-ized characteristic boundary conditions (Kim & Lee, 2000) and characteristic interfaceconditions (Kim & Lee, 2003).1 Governing equations in cylindrical coordinatesThe starting point are the compressible Navier{Stokes equations in cylindrical coordi-nates, given as The °ow is assumed to be an ideal gas with constant speci¯c heat coe±-cients and all quantities were made dimensionless using the °ow quantities at a referencelocation in the °ow (the free-stream/in°ow location was used). The radius of the bodyis chosen as the reference length. The non-dimensionalization results in the followingdimensionless parameters:Re =½¤1u¤1r¤¹¤1; M =u¤1a¤1; Pr =¹¤1c¤p·¤1:With z;r;µ denoting the streamwise, radial and azimuthal directions, respectively, andu;v;w denoting the velocity components in z;r;µ directions, respectively, the non-dimensionalcompressible Navier{Stokes equations in cylindrical coordinates are@Q@t+@E@z+@F@r+1r@G@µ= Sv ; (1)where the conservative variable and the inviscid °ux vectors areQ = [½; ½u; ½v; ½w; ½E]TE = [½u; ½uu + p; ½uv; ½uw; u(½E + p)]TF = [½v; ½uv; ½vv + p; ½vw; v(½E + p)]TG = [½w; ½uw; ½vw; ½ww + p; w(½E + p)]T ;with the total energy de¯ned as E = T=[°(° ¡ 1)M2] + 1=2uiui and ° = 1:4. The sourceterm Sv consists of the viscous °ux derivatives and the additional terms obtained in thetransformation to cylindrical coordinates (H)Sv =@Ev@z+@Fv@r+1r@Gv@µ¡1rH ; (2)withEv = [0; ¿zz; ¿zr; ¿zµ; ¡qz + u¿zz + v¿zr + w¿zµ]TFv = [0; ¿rz; ¿rr; ¿rµ; ¡qr + u¿rz + v¿rr + w¿rµ]TGv = [0; ¿µz; ¿µr; ¿µµ; ¡qµ + u¿µz + v¿µr + w¿µµ]TH = [½v; ½uv ¡ ¿zr; ½vv ¡ ½ww ¡ ¿rr + ¿µµ; 2½vw ¡ 2¿µr;v(½E + p) + qr ¡ u¿rz ¡ v¿rr ¡ w¿rµ]T :1The molecular stress tensor components are¿zz =2¹3Re"2@u@z¡@v@r¡1rÃ@w@µ+ v!#¿rr =2¹3Re"¡@u@z+ 2@v@r¡1rÃ@w@µ+ v!#¿µµ =2¹3Re"¡@u@z¡@v@r+ 21rÃ@w@µ+ v!#¿rz =¹Re"@u@r+@v@z#¿µz =¹Re"@w@z+1r@u@µ#¿µr =¹Re"1rÃ@v@µ¡ w!+@w@r#:The heat-°ux vector components areqz =¡¹Pr(° ¡ 1)M2Re@T@zqr =¡¹Pr(° ¡ 1)M2Re@T@rqµ =¡¹Pr(° ¡ 1)M2Re1r@T@µ;where the Prandtl number is assumed to be constant at Pr = 0:72. The molecularviscosity ¹ is computed using Sutherland's law (c.f. White, 1991), setting the ratio ofthe Sutherland constant over freestream temperature to 0:36867. To close the systemof equations, the pressure is obtained from the non-dimensional equation of state p =(½T)=(°M2).2 Transformation to general cylindrical coordinatesIn order to allow for complex geometries, the (z;r) coordinate system is mapped togeneral coordinates (»;´). By deliberately restricting the coordinate mapping to two-dimensions, the azimuthal grid is required to be equidistant. What might seem as aprofound constraint actually is advantageous in two ways: ¯rstly an e±cient Fourierspectral method easily can be employed for the discretization of the µ-direction, andsecondly, a smaller number of metric terms is required than for the most general case, andall metric terms needed in the current case are two-dimensional (versus three-dimensionalif a mapping was performed in all coordinate directions). Therefore much less memory isrequired for the simulations. The streamwise and radial derivatives need to be expressedin terms of the new variables. By using the chain rule, the following expressions can bederived@@z=1J"@r@´@@»¡@r@»@@´#= r¤´@@»¡ r¤»@@´; (3)2@@r=1J"¡@z@´@@»+@z@»@@´#= z¤»@@´¡ z¤´@@»; (4)where J = z»r´¡z´r» is the determinant of the coordinate transformation. For conciseness,the metric terms are abbreviated as, e.g., r´ = @r@´, and the asterisk denotes a metric termalready divided by J.Transforming the original equation (1) to a non-conservation form in generalized coordi-nates using the chain rule, we obtain@Q@t+Ã»z@E@»+ »r@F@»!+Ã´z@E@´+ ´r@F@´!+1r@G@µ= Sv : (5)This non-conservative form can be transformed into a characteristic form in the directionnormal to a computational boundary (either the boundaries of the domain or interfacesbetween blocks or sub-domains). Considering the » direction, the quasi-linear character-istic wave equation is@R@t+ ¤@R@»= ¡P¡1S¤v ; (6)where an underscore denotes a matrix. Only the °uxes in the » direction are considered onthe right-hand-side of equation (6), while all °uxes in the other directions are containedin the modi¯ed source termS¤v = Sv ¡Ã´z@E@´+ ´r@F@´!¡1r@G@µ= Sv ¡Ã¡r¤»@E@´+ z¤»@F@´!¡1r@G@µ: (7)For a de¯nition of the characteristic R variables and the transformation matrix from theconservative to the characteristic variables, the reader is referred to (Kim & Lee, 2000)and the Appendix. The eigenvalues in ¤ are¤ =0B BB BBB B@UUUU + cq»2z + »2rU ¡ cq»2z + »2r1C CC CCC CA=0BB BB BBB BB@UUUU + cr³r¤´´2+³z¤´´2U ¡ cr³r¤´´2+³z¤´´21CC CC CCC CCA; (8)where c is the speed of sound and the contravariant velocity is given byU = »zu + »rv = r¤´u ¡ z¤´v : (9)Following (Kim & Lee, 2000), the convection terms in (6) are transformed to the °uxderivatives of (5) as»z@E@»+ »r@F@»= r¤´@E@»¡ z¤´@F@»= P ¤@R@»= PL ; (10)thus the convection terms are de¯ned asLi = ¸i@Ri@»(i = 1;2;:::;5) : (11)3The transformation matrix from characteristic to conservative variables is given in (Kim& Lee, 2000), and in modi¯ed form for the current case in the Appendix.The ¯nal step is to derive the conservation form of the Navier{Stokes equations in gen-eralized coordinates. Following Anderson (1995), the r and z derivatives of equation (1)can be transformed as follows@^ Q@t+@^ E@»+@^ F@´+Jr@G@µ= ^ Sv ; (12)where the inviscid °ux vectors in generalized coordinates are now given as^ Q = JQ ; ^ E = r´E ¡ z´F ; ^ F = ¡r»E + z»F ;and ^ Sv is the source vector that contains the viscous °uxes in generalized coordinates andall non-conservative terms due to the cylindrical coordinate system^ Sv =@^ Ev@»+@^ Fv@´+Jr@Gv@µ¡JrH ; (13)with^ Ev = r´Ev ¡ z´Fv ; ^ Fv = ¡r»Ev + z»Fv :3 Characteristic equations in »-directionWhen using the conservative formulation of the Navier{Stokes equations in generalizedcoordinates as basis, the quasi-linear characteristic wave equation (for the »-direction)can now be written as@R@t+ ¤@R@»= Sc ; (14)where the modi¯ed source term isSc =1JP¡1(^ Sv ¡"E@r´@»¡ F@z´@»+@^ F@´+Jr@G@µ#)=1JP¡1(@^ Q@t+@^ E@»¡ E@r´@»+ F@z´@»): (15)The characteristic convection term on a boundary or interface can now be computed asL =1JP¡1(@^ E@»¡ E@r´@»+ F@z´@»): (16)The characteristic convection term is then either corrected by imposing interface condi-tions according to (Kim & Lee, 2003), using the modi¯ed source term from (15), or usingthe appropriate LODI relations given in (Kim & Lee, 2000). The normal-°ux terms are¯nally recalculated using the corrected convection terms L¤i according toÃ@^ E@»!¤= JPL¤ + E@r´@»¡ F@z´@»: (17)44 Characteristic equations in ´-directionThe quasi-linear characteristic wave equation for the ´-direction can be written as@R´@t+ ¤@R´@´= Sc´ ; (18)where the modi¯ed source term isSc´ =1JP¡1´(^ Sv ¡"¡E@r»@´+ F@z»@´+@^ E@»+Jr@G@µ#)=1JP¡1´(@^ Q@t+@^ F@´+ E@r»@´¡ F@z»@´): (19)The characteristic convection term on a boundary or interface can now be computed asL =1JP¡1´(@^ F@´+ E@r»@´¡ F@z»@´): (20)The characteristic convection term is then either corrected by imposing interface condi-tions according to (Kim & Lee, 2003), using the modi¯ed source term from (15), or usingthe appropriate LODI relations given in (Kim & Lee, 2000). The normal-°ux terms are¯nally recalculated using the corrected convection terms L¤i according toÃ@^ F@´!¤= JP´L¤ ¡ E@r»@´+ F@z»@´: (21)5 ImplementationIn order to advance the solution in time, the right-hand-side of equation 12 needs to besolvedRHS =@^ Q@t= ¡"@^ E@»+@^ F@´+Jr@G@µ+JrH ¡@^ Ev@»¡@^ Fv@´¡Jr@Gv@µ#: (22)The advantage of solving for ^ Q is that all conservative variables are known in (the equidis-tant) computational space. Thus, if ¯ltering is required, ¯lters derived for equidistantlyspaced grids can be employed without loss of accuracy.To make the algorithm of the novel code most e±cient, the number of three-dimensionalarrays to be stored needs to be minimized. At the same time, the number of arithmeticoperations needs to be kept at bay, hence a good compromise between the number ofstored quantities and variables to be recomputed needs to be found.ReferencesAnderson, J. 1995 Computational Fluid Dynamics, The Basics with Applications.McGraw-Hill.5Kim, J. & Lee, D. 2000 Generalized characteristic boundary conditions for computa-tional aeroacoustics. AIAA J. 38 (11), 2040{2049.Kim, J. & Lee, D. 2003 Characteristic Interface Conditions for Multiblock High-OrderComputation on Singular Structured Grid. AIAA Journal 41 (12), 2341{2348.White, F. M. 1991 Viscous Fluid Flow. McGraw Hill.AppendixThe transformation matrices between the conservative and the characteristic variables,and vice versa, are given asP¡1 =0B BBB BB BBB@B0 ¢ lz (° ¡ 1) uc2 ~ »z (° ¡ 1) vc2 ~ »z (° ¡ 1) wc2 ~ »z ¡~ »r½ ¡°¡1c2 ~ »zB0 ¢ lr (° ¡ 1) uc2 ~ »r (° ¡ 1) vc2 ~ »r (° ¡ 1) wc2 ~ »r +~ »z½ ¡°¡1c2 ~ »rB0 ¢ lµ~ »r½ ¡~ »z½ 0 0c½³°¡12 M2 ¡v¢l»c´C+ ¢ lz C+ ¢ lr C+ ¢ lµ°¡1½cc½³°¡12 M2 +v¢l»c´C¡ ¢ lz C¡ ¢ lr C¡ ¢ lµ°¡1½c1C CCC CC CCCA;(23)withB0 = f1 ¡ [(° ¡ 1)=2]M2gl» ¡ (1=½)(v £ l») ;C§ = §(l»=½) ¡ [(° ¡ 1)=½c]v ; v = (u;v;w)Tl» = (~ »z; ~ »r;0) =µ1=q»2z + »2r¶(»z;»r;0)=Ã1=r³r¤´´2+³z¤´´2!³r¤´;¡z¤´;0´;lz = (1;0;0) ; lr = (0;1;0) ; lµ = (0;0;1) :P =0BB BB BB@~ »z ~ »r 0½2c½2cu~ »z u~ »r ½~ »r½2c(u + ~ »zc)½2c(u ¡ ~ »zc)v~ »z v~ »r ¡½~ »z½2c(v + ~ »rc)½2c(v ¡ ~ »rc)w~ »z ¡ ½~ »r w~ »r + ½~ »z 0½2cw½2cwb ¢ lz b ¢ lr b ¢ lµ½2c(H + cv ¢ l»)½2c(H ¡ cv ¢ l»)1CC CC CCA; (24)withb =jvj22l» + ½(v £ l») ; H =jvj22+c2° ¡ 1:For the current case, where mapping is only performed in the z¡r-plane and with ~ »z = lr¤´and ~ »r = ¡lz¤´ withl =1r³r¤´´2+³z¤´´2 ; (25)6and using °1 = ° ¡ 1 and q = u2+v2+w22 , the transformation matrices becomeP¡1 =0BB BBB BB BB@³1 ¡°12 M2´lr¤´ ¡ l½wz¤´°1uc2 lr¤´°1vc2 lr¤´°1wc2 lr¤´ +lz¤´½ ¡°1c2lr¤´ ³°12 M2 ¡ 1´lz¤´ ¡ l½wr¤´ ¡°1uc2 lz¤´ ¡°1vc2 lz¤´ ¡°1wc2 lz¤´ +lr¤´½°1c2lz¤´l½³uz¤´ + vr¤´´¡lz¤´½ ¡lr¤´½ 0 0c½³°12 M2 ¡ lUc´l½r¤´ ¡°1½cu ¡ l½z¤´ ¡°1½cv ¡°1½cw°1½cc½³°12 M2 + lUc´¡ l½r¤´ ¡°1½cu l½z¤´ ¡°1½cv ¡°1½cw°1½c1CC CCC CC CCA;(26)P =0BB BBB B@lr¤´ ¡lz¤´ 0½2c½2culr¤´ ¡ulz¤´ ¡½lz¤´½2c(u + lr¤´c)½2c(u ¡ lr¤´c)vlr¤´ ¡vlz¤´ ¡½lr¤´½2c(v ¡ lz¤´c)½2c(v + lz¤´c)wlr¤´ + ½lz¤´ ¡wlz¤´ + ½lr¤´ 0½2cw½2cwqlr¤´ + ½lwz¤´ ¡qlz¤´ + ½lwr¤´ ¡½l³uz¤´ + vr¤´´½q2c +½c2°1 +½l2 U½q2c +½c2°1 ¡½l2 U1CC CCC CA;(27)P¡1´ =0B BB BBB BB B@³°12 M2 ¡ 1´l´r¤» +l´½ wz¤» ¡°1uc2 l´r¤» ¡°1vc2 l´r¤» ¡°1wc2 l´r¤» ¡l´z¤»½°1c2l´r¤» ³1 ¡°12 M2´l´z¤» +l´½ wr¤»°1uc2 l´z¤»°1vc2 l´z¤»°1wc2 l´z¤» ¡l´r¤»½ ¡°1c2l´z¤»¡l´½³uz¤» + vr¤»´ l´z¤»½l´r¤»½ 0 0c½³°12 M2 ¡l´Vc´¡l´½ r¤» ¡°1½cul´½ z¤» ¡°1½cv ¡°1½cw°1½cc½³°12 M2 +l´Vc´l´½ r¤» ¡°1½cu ¡l´½ z¤» ¡°1½cv ¡°1½cw°1½c1C CC CCC CC CA;(28)P´ =0BBB BB B@¡l´r¤» l´z¤» 0½2c½2c¡ul´r¤» ul´z¤» ½l´z¤»½2c(u ¡ l´r¤»c)½2c(u + l´r¤»c)¡vl´r¤» vl´z¤» ½l´r¤»½2c(v + l´z¤»c)½2c(v ¡ l´z¤»c)¡wl´r¤» ¡ ½l´z¤» wl´z¤» ¡ ½l´r¤» 0½2cw½2cw¡ql´r¤» ¡ ½l´wz¤» ql´z¤» ¡ ½l´wr¤» ½l´³uz¤» + vr¤»´½q2c +½c2°1 +½l´2 V½q2c +½c2°1 ¡½l´2 V1CCC CC CA;(29)where M2 =2qc2, U = ur¤´ ¡ vz¤´, V = vz¤» ¡ ur¤», andl´ =1r³r¤»´2+³z¤»´2 ; (30).7

Generalized cylindrical coordinates for characteristic boundary conditions and characteristic interface conditions

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