Foundations of Fluid Mechanics I
Foundations of Fluid Mechanics I M E 521
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Date Created: 11/01/15
Incompressible 3D Laminar Flow Boundary Layer Equations Author John M Cimbala Penn State University Latest revision 17 January 2008 AssumptionsApproximations o x and z coordinates lie along the body surface y coordinate is normal to the body surface 0 The uid is either an incompressible liquid or a nearly incompressible ideal gas at very low Mach number 0 Gravity is neglected Scale Factors or Stretching Factors De ne is the distance from the fixed origin to a point inside the boundary layer r is the distance from the xed origin to the body surface and a is the unit outward normal in the direction awa from the bod surface C t39 39t 7 4 4539 as on 1nu1 y m lbf 4 rig xmomentum W w m v nl m7 g sn em y mom entum A r y A n momentum a t g w aw Mm Euler Equations for the Irrotational Outer Flow U and Walong the body surface Note These expressions can replace the pressure gradient terms in the x and Zmomentum equations above so that the boundary layer equations can be written in terms of the known outer ow velocity eld at the wall Uxz and Wxz It m7 can as a sir e x momentum 4Lvse 7 Eiffel g W72 3 z momentum a I f39 12 3 52 if 653 11162 Boundary Conditions N0 slip conditions it v w 0 at y 0 at the surface of the body for all x and Z Edge conditions it Uand w Was y a 00 outside the BL for all x and Z 0 Starting conditions Must specify u and w pro les at some x and 2 locations starting pro les to begin the calculations Reynolds Stress Turbulence Models SecondOrder Closure Author John M Cimbala Penn State University Latest revision 14 April 2008 The Exact Reynolds Stress Transport Equation Consider incompressible turbulent flow without gravity The mean continuity and momentum equations reduce to 0U taU 0P 039 0U 0 l and p 39 U 11 7 pufu 2 In add1t1on to these 4 equat1ons and 4 pr1mary r at 026 6x1 0x 0x unknowns U andP which are functions of x and t The Reynolds stress tensor in the momentum equation adds six additional 39 1 unknowns Exact transport equations for the Reynolds stress tensor can 3e derived 0 or 0U 0U 69 07 Uk i rM W rm Aj v M 61 8 0t xk 026k Oxk Oxk Oxk l II III IV V VI 1 Total rate of change of Reynolds stress following a fluid particle unsteady plus advective terms 3 ll Production of Reynolds stress often given the symbol Py Ill Pressurestrain correlation tensor a rediszribution term often called the pressurestrain redistribution term and sometimes i 0 0quot g1ven the symbol in or qjj 1nstead Here Hy E p L Jr 0x 0x lV Spatial transport diffusion of Reynolds stress byimolecular viscous effects V Spatial transport diffusion of Re nolds stress b turbulent uctuations Here C1 jk sometimes calledDUk instead is the diffusion correlation defined as CVC E puxujuk pu15k VI Rate of viscous dissipation of Reynolds stress 81 E 21 The Modeled Reynolds Stress Transport Equation Terms 1 H and IV are exact However terms Ill V and VI need to be modeled Let s consider each of these separately I Term III This is the hardest term to model and has received the most attention It is typically split into slow and rapid BUk parts UV Ay MW axl plus a wallreflection effect or pressureecho effect near solid walls The slow pressure 5016 There are many models of the rapid and wallreflection terms the most Vania 8 strain term IS modeled as A m C1 239 K popular of which is that of Launder Reece and Rodi LRR given bel w See Wilcox 1998 for others K 01 071k 07y I Term V Th1s term IS modeled by the gradient di usion approximation CC E CS quot b 39 quot quot b 0x 0x Oxk I Term VI Since most of the viscous dissipation occurs at the smallest scales which are nearly isotropic most modelers assume Since sis another unknown in the problem a seventh additional transport equation for 8 must be solved 7 generally similar to that used in the K gmodel A complete example the Laun derReeeeRodi model is provided below n is the normal distance from the nearest wall or or a K2 or or or 2 0U 0U 1m Uk 1m 7P1 7 y CS 1km km 1me p551 PU le J TJk x at 0xk 0xk 8 0x1 0x 0xk 3 b 0xk b 0xk 3 g A A A g Kr Hy C1 EUM PK5U Py P5y 51 P5y 7PKEy iEa ij0125ETm PK5V 1015ij 11 0U Uk A A w1th Dy 23km 5 11km E or 8C2l l 3 8C22l l 7 60C2455 C1 l8 C2 060 C 011 We also need an 1 x 08 08 s 0U 0 K 08 82 equation for g p pUk C T iCap with Cs018 C l44 and C 192 g m ax K 5 T 0t Oxk a K m Oxk 50x