A Method of Solving Compressible Navier Stokes Equations in Cylindrical Coordinates Using Geometric Algebra

Transcription

1 Pepints NOT PEER-REVIEWED Posted: 8 Decembe 08 Aticle A Method of Solving ompessible Navie Stokes Equations in ylindical oodinates Using Geometic Algeba Tey E. Moschandeou,, * London Intenational Academy, 365 Richmond Steet London, ON, N6A 3, anada; Depatment of Applied Mathematics, Faculty of Science, Westen Univesity, London Ontaio, N6A 5, anada * oespondence: TMoschandeou@lia-edu.ca o tmoschan@uwo.ca; Tel.: Abstact A method of solution to solve the compessible unsteady 3D Navie-Stokes Equations in cylindical co-odinates coupled to the continuity equation in cylindical coodinates is pesented in tems of an additive solution of the thee pinciple diections in the adial, azimuthal and z diections of flow. A dimensionless paamete is intoduced wheeby in the lage limit case a method of solution is sought fo in the bounday laye of the tube. A eduction to a single patial diffeential equation is possible and integal calculus methods ae applied fo the case of a body foce diected to the cente of the tube to obtain an integal fom of the Hunte-Saxton equation. Also an extension fo a moe geneal body foce is shown whee in addition thee is a otational foce applied.. Intoduction ompessible flow has many applications some of which ae of physics, mathematics and engineeing inteest. In geneal we have two types of flows, intenal and extenal. Intenal flows in ducts ae impotant in industy and nozzles and diffuses used in engines ae also an applied aea fo these types of flows. In geneal, density changes ae elated to tempeatue changes. Extenal flows can be impotant fo aiplanes and pojectiles whee compessiblilty effects ae impotant. The Navie Stokes equations have been dealt with extensively in the liteatue fo both analytical 8] and numeical solutions 9], 3]. Some wok conveting the compessible Navie-Stokes equations to the Schödinge equation in quantum mechanics by means of tansfomations has been caied out by Vadasz. 0]. Geneal mathematical and computational methods fo compessible flow have been outlined in 6]. Methods in moe geneal fluid mechanics ae also addessed in ]. In the context of functional analysis it has been shown in 5] that geneally the motion of a compessible fluid with fixed initial velocity field and constant initial density conveges to that of an incompessible fluid as it s sound speed goes to infinity. Some analytical methods such as in ] have been successfully caied out fo one and two dimensional isentopic unsteady compessible flow. Little is known though fo analytical methods fo thee dimensional compessible unsteady flow fo both isentopic and non-isentopic flow. In the pesent wok I intoduce a new pocedue to wite the compessible unsteady Navie Stokes equations with a geneal spatial and tempoal vaying density tem in tems of an additive solution of the thee pinciple diections in the adial, azimuthal and z diections of flow. A dimensionless paamete is intoduced wheeby in the lage limit case a method of solution is sought fo in the bounday laye of the tube. It is concluded that the total divegence of the flow can be expessed as the integal with espect to time of the line integal of the dot poduct of inetial and azimuthal velocity. The line integal is evaluated on a contou that is annula and taces the bounday laye as time inceases in the flow. A eduction to a single patial diffeential equation is possible and integal calculus methods ae applied fo the case of a body foce diected to the cente of the tube to obtain an integal fom of the Hunte-Saxton equation. Also an extension fo a moe geneal body foce is shown whee in addition thee is a otational foce applied. 08 by the authos. Distibuted unde a eative ommons BY license.

2 Pepints NOT PEER-REVIEWED Posted: 8 Decembe 08 of 37. A new composite velocity fomulation The 3D compessible cylindical unsteady Navie-Stokes equations ae witten in expanded fom, fo each component, u,u θ and u z : u + u u + u θ u θ u θ + u z u z ρ u + u u + u θ + u z + ρ + u + u θ u θ θ + u z p Fg = 0 u θ + u u θ + u θ u θ θ + u u u θ + u θ z z ρ u θ θ + u θ θ + u θ z u θ + u θ + ρ + u θ + u θ θ + u θ + u θ z p θ Fg θ = 0 u z + u u z + u θ u z z u z z ρ u z θ + u z u z + u z θ + u z z + + ρ u z + u z θ + u z z p z Fg z = 0 3 whee u is the adial component of velocity, u θ is the azimuthal component and u z is the velocity component in the diection along tube, ρ is density, is dynamic viscosity,fg, Fg θ, Fg z ae body foces on fluid. The total gavity foce vecto is expessed as FT = Fg, Fg θ, Fg z. The following elationships between staed and non-staed dimensional quantities togethe with a non-dimensional quantity δ ae used: u = δ u 4 u θ = δ u θ 5 u z = δ u z 6 = δ 7 θ = θ 8

3 Pepints NOT PEER-REVIEWED Posted: 8 Decembe 08 3 of whee δ = z = δ z, t = t 9 ρ ρ θ + ρ as pat of the continuity equation Eq. The egion of inteest is confined to within the bounday laye which is ceated in an assumed compessible viscous flow in the tube. Using Eqs.4-9,multiplying Eqs-3 by catesian unit vectos e =, 0, 0, e θ = 0,, 0 and k = 0, 0, espectively and adding Equations -3 gives the following equation, fo the esulting composite vecto L = δ u e + δ u θ e θ + δ u z k, δ L δ + u L δ + u θ L δ θ + L δ u z δ u θ e + δ 3 u u θ e θ δ L ρ + δ L + L δ + L δ θ + δ u δ θ e θ u θ δ θ e + δ L δ u + u θ + u u z + u z θ + u z e δ p k + δ ρ e + δ ρ u θ θ + u θ θ + u θ e θ p θ e θ + δ p ρ k FT = 0 0 Futhemoe one can take the coss poduct of Equation 0 with e and e θ to fom two equations espectively whee in the second case multiplication by δ to cancel squaed nonlinea tems, δ u θ e, δ 3 u u θ e θ, leads to, δ δ LA + u LA δ + u θ LA δ θ + LA δ u z + δ u θ δ L A ρ + δ LA + LA δ + LA δ δ + δ u θ θ + u θ θ u + u θ + u δ ρ + u θ k + δ θ + δ k + δ ρl ρ δ u θ k δ u z + u z u z + u z θ + u z u ρ z ρ k u θ θ k + δ LA + θ + u z e θ e δ ρ p θ k p e θ + p δ ρ k δ p ρ e + FT = whee LA = u z e δ u z eθ + δ u u θ k, LA = L e + δ L e θ and continuity equation, Eq below has been used. Equation 0 is sufficient to cay out the subsequent analysis A Solution Pocedue fo δ abitaily lage in quantity Multiplication of Eq.0 by ρ δ and Eq. below by L δ, addition of the esulting equations 7], and using the odinay poduct ule of diffeential multivaiable calculus a fom as in Eq.3 is obtained wheeby I set a = ρ L.

4 Pepints NOT PEER-REVIEWED Posted: 8 Decembe 08 4 of The continuity equation in cylindical co-odinates is ρ + ρ δ u + u θ ρ θ + ρ u z = ρ δ u + δ u + u θ θ ρ a + a a + ρ b b = δ b + 3 b whee δ = δ. I denote it by below. + u z Taking the geometic poduct in the pevious equation with the inetial vecto tem, = ρl + P + FT 3 f = a a whee b = a ρ is defined, whee in the context of Geometic Algeba, the following scala and vecto gade equations aise, f ρ b + ρ ρ b + f + b f ρ b = f b + 3 f b + f P 5 b + ρ b ρ + bρ b = b + 3 b + P + FT 6 The geometic poduct of two vectos 4], is defined by A B = A B + A B. Taking the divegence of Eq.6 esults in b + ρ ρ b + ρ b ρ + bρ b = b + 3 b + P + FT 7 Upon multiplication of Eq7 by, the esulting equation is H = ρ3 b f ρ 8 H b + ρ b f b + H ρ b ρ + H bρ b = H b + 3 H b + H P + H FT 9

5 Pepints NOT PEER-REVIEWED Posted: 8 Decembe 08 5 of which esults upon using Eq5 in, H b f ρ b + ρ ρ b f f b 3 f b f P + H b ρ ρ + H bρ b = H b + 3 H b + H P + H FT 0 The continuity equation is witten in tems of b as, and ρ + ρ b + ρ b ρ + ρ ρ b = 0 whee the following compact expession is given, b = ρ ρ ρ ρ b Y = b 3 Multiplying by f f in Eq.0 and, using popeties of thid deivatives involving the gadient and in paticula the fact that the Laplacian of the divegence of a vecto field is equivalent to the divegence of the Laplacian of a vecto field, leads to the following fom, W Y Gρ, ρ W Fρ, ρ b f + f P VW Y b ρ ρ 3 f b f f + Ω b ρ ρ 3 f f f b b ρ ρ 3 f f 3 f b b f f b f bρ b + P + FT = 0 4 whee Ω = H b ρ ρ in Eq.4 and, W = f b f f b = f b f f b + f b f f b = ξ + f f b f b 5 This involves the vecto pojection of b onto f which is witten in the conventional fom, poj f b = f b f f 6 Equation 4 can be witten compactly as Y Gρ, ρ V Y P = U f Qρ, ρ, b, b, P, FT + f ] U f b 7

6 Pepints NOT PEER-REVIEWED Posted: 8 Decembe 08 6 of whee U f ξ is the scala pojection fo b, Q and hence fo a constant α, Y Gρ, ρ Qρ, ρ V Y P =, b, b, P, FT + f b = α 8 with solution in tems of a function B, Y = b = Bα,, θ, z, t 9 If the ate of change of ρ with espect to t is changing exponentially in time as Bθ, ze ct fo some eal constant c such that Oc 0, B a geneal function which vanishes in Gρ, ρ = ρ, and since ρ ρ is O/ in bounday laye, it may be poven that B is witten linealy in tems of a sepaable function, that is Bα,, θ, z, t = F F θ F 3 z F 4 t /4 + ln + 3 c + α 30 3 whee F i ae the sepaable pats of the function and j ae abitay constants. Also it is assumed that highe ode deivatives ae negligible in the bounday laye, so that the Laplacian of pessue is dopped and the gadient of pessue is peseved. Fo c appoaching zeo and using the popeties of the nom, Y, θ, z, t F F θ F3 z F4 t ] b M Q ρ, ρ, b b,, P, FT + f = 0 3 Futhemoe using the popety fo a nom on a vecto space of continuous functionssince away fom = 0, A = 0 iff A = 0, and Eq.6 and the second line of Eq.4 gives, Y, θ, z, t F F θ F3 z F4 t ] b Kρ, ρ M b ρ b ρ ρ ] Fρ, f + P b bρ b + FT = 0 3 whee M is the function on the vey ight side of Eq.30. ] b F F θ F3 z F4 t b ] Kρ, ρ M b ρ b ρ ρ Fρ, f + P b FT = 0 33 whee Fρ, ρ = ρ 3 ρ. Ω in Eq.4 vanishes due to assumption on ate of change of density with espect to t and c 0, and finally Eq. has been used with a calculation done to show that the expession educes in going fom Eq 3 to Eq33. This occus in the bounday laye due to density being O/. In the bounday laye it can be poven that F is decaying as a J 0 Bessel function and K is

7 Pepints NOT PEER-REVIEWED Posted: 8 Decembe 08 7 of an inceasing function adially thee with the following esulting upon dividing by K and M, ] b ρ b ρ ρ Fρ, f + P b FT = 0 34 Dividing by Fρ, ρ and taking the cul of Eq.34 simplifies to the following in the bounday laye, F b f F b FT = 0 35 The inetial vecto f = f / a a. Multiply Eq.35 by the nomal vecto cosθ a which is the nomal component of a at wall of moving contol volume V in Figue, cosθ a F ] b f F b FT = 0 36 hoosing ρ so that F = the following is used, Recalling Divegence theoem and Stoke s theoem, fo geneal F, FdV = F ˆndS V SV S F ds = F d 37 whee = defines the union of two contous whee the fist is at the bounday laye edge and the othe is along the wall Fig. and S consists of all sufaces of contol volume. Defining the following vecto field, W = b b FT, 38 SV W ˆndS = WdV 39 V = whee ˆn = cosθ a, and Stoke s theoem has been used. f d 40 f Tds = b + b FT d 4

8 Pepints NOT PEER-REVIEWED Posted: 8 Decembe 08 8 of whee T is unit tangent vecto to closed cuve and ds is ac length, f + b + b FT d = 0 4 The thid tem in the paenthesis in Eq4 is integated by pats fo line integal and I obtain the following, f + b F T b d = 0 43 Paametizing the cicle as = gθ in pola coodinates it can be poven that the line integal in Eq43 is, f + b F T b dθ f + b F T b d = 0 44 The nomal fom of Geen s theoem can be used fo the line integal in Eq44,setting fist, M = f + b F T b 45 The line integal in Eq44 is equal to the following, N = f + b F T b 46 R M + N ddθ 47 θ whee M and N ae given by Eqs45 and 46 espectively and R is the annulus with bounday. The extenal foce F T is expessed as follows.see Appendix, 46 It follows that Eq.47 becomes, ˆ b F T = sinθ d 48 BL θ 47 R sinθ ˆ BL b θ d b + sinθ θ b + f + b θ θ ddθ 49 and since the bounday laye is thin the fist integal vanishes and the Eq.47 becomes, R sinθ ] b + f + b θ θ ddθ 50 In the bounday laye the function b = b θ, t and b = b, t and thus, ˆ π 0 BL b b θ + b t θ sinθ ln b θ dθ 5 BL

9 Pepints NOT PEER-REVIEWED Posted: 8 Decembe 08 9 of Dividing by BL,fo θ π/, and using L Hopital s ule, I obtain the Hunte Saxton equation with BL = The Hunte-Saxton Equation Substituting f and F T into Eq45 and 46 since F T = 0 and f, b tems ae negligible in the bounday layerecall u = u /δ, t = t /δ then one can obtain an integal fom of the Hunte-Saxton equation using Eq.47, b t + b b θ θ = b θ It necessaily follows that fo small θ appoximation that b 0,t θ = 0, deivatives at zeo ae zeo. b 0,t θ 5 = 0, and highe It is of inteest that a moe complicated patial diffeential equation is manifest upon taking F T to be non zeo and thus defining a otational foce elated to the voticity of the fluid elements in the bounday laye. Figue. ompessible Viscous Flow in a Tube, z bl is the distance to achieve maximum bounday laye height. Figue. A Typical set of contou lines between edge of bounday laye and tube wall In the inteval 0, t] the bounday laye stats to fom at t = 0 and eaches maximum height at time t. 60 See Fig. fo contol volume ove the gowth of the bounday laye. The time dependence is shown in 6 the inetial tem f. The ight side of Eq.40 consists of nonlinea inetial tem f and Eq.39 shows the 6 dependence on these gadients and ate of change of cul of b with espect to t onclusion An attempt has been made to educe the compessible Navie Stokes equations coupled to the continuity equation in cylindical co-odinates to a simple poblem in tems of an additive solution of the thee pinciple diections in the adial, azimuthal and z diections of flow. A dimensionless

10 Pepints NOT PEER-REVIEWED Posted: 8 Decembe 08 0 of Figue 3. Vectos a, c, d as well as d and d which ae pointing in the diection of inceasing gavitational foce.appendix paamete is intoduced wheeby in the lage limit case a method of solution is sought fo in the bounday laye of the tube. By seeking an additive solution using the continuity equation and the simplified vecto equation obtained by a simila pocedue and using the poduct ule of diffeentiable calculus, and using Geometic Algeba as a stating point, it follows though analysis that the integal of total divegence of a specific vecto field ove time can be expessed as the integal with espect to time of the line integal of the dot poduct of inetial and azimuthal velocity. The line integal is evaluated on a contou that is annula and taces the bounday laye as time inceases in the flow. It has been shown that a eduction of the 3D compessible unsteady Navie-Stokes equations to a single patial diffeential equation is possible and integal calculus methods ae applied fo the case of a body foce diected to the cente of the tube to obtain an integal fom of the Hunte-Saxton equation. Also an extension fo a moe geneal body foce has been shown whee in addition thee is a otational foce applied. 6. Appendix Refeing to Fig.3, let, d = k a 53 d = m c 54 It follows that, d + d = d = k a + m c 55 If a and c ae in the same diection then km > 0 and k, m ae eithe both positive o both negative. Also accoding to Fig.3 it follows that, d = c + d 56 d = a + d 57 If the cicle shinks to an abitay small cicle in diamete then d d 58

11 Pepints NOT PEER-REVIEWED Posted: 8 Decembe 08 of d d 59 Summing up all the d i components and letting b i e θ epesent the tangential velocity fo each point on the cicumfeence of disk R, it follows fo N distinct points that, N N d i = b i 60 i= i= Thus in the limit as the cicle diamete gets small, intoduce a function g defined as, g = b θ whee b = b e θ and b θ is the vecto pointing towads the cente of the disk in R. Acknowledgment G.K. Batchelo, An intoduction to fluid dynamics, ambidge Univesity Pess, R. ai, Some Explicit analytical solutions of unsteady compessible flow, tansactions of the ASME, Jounal of fluids engineeing, Vol. 0, 998, A.J. hoin, The numeical solution of the Navie-Stokes equations fo an incompessible fluid, Bull. Ame. Math Soc. Volume 73 No.6, 967, Doan, A. Lasenby, Geometic Algeba fo Physicists, ambidge Univesity Pess, D.G. Ebin, Motion of a slightly compessible fluid, Poc. Nat. Acad. Sci USA, Vol 7,No., 975, M. Feistaue, J. Felcman and I. Staskaba, Mathematical and computational methods fo compessible flow, laendon Pess-Oxfod Science Publications, T. Moschandeou, A New Analytical Pocedue to Solve Two Phase Flow in Tubes, Math. omput. Appl. 08, 3, 6. doi: /mca L. M. Peeia, J. S. Peez Gueeo and R. M. otta, Integal tansfomation of the Navie-Stokes equations in cylindical geomety, omputational Mechanics, Volume, Issue, Febuay 998, Taylo, P.Hood, A numeical solution of the Navie-Stokes equations using the finite element technique, omputes and Fluids, Volume Issue, 973, P. Vadasz, Rendeing the Navie-Stokes equations fo a compessible fluid into the Schödinge Equation fo quantum mechanics, Fluids 06.