FSU EML 3016 - Basic Equations - a differential approach

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Governing Equations in Differential FormContinuity EquationPhysical InterpretationExample (suction boundary layer control)Example (cont.)Governing Equations in Differential FormVery often, we would like to examine the detailed variation of a fluid flow field instead of just evaluating the integral effects. For example, local flow behavior near the solid surface determines both the convective heat transfer and the skin friction between the surface and the fluid.In these situations, governing equations in a differentil form are needed:The integral formula such as the massCS conservation equation: V dA 0 is still valid. However, it has to be evaluated inside an infinitesimal element in order to be able to predict the local flow behavior.CVdt     rr• Take the limit such that the control volume approach to infinitesimal small: consequently, all fluid properties within the volume can be considered constant.• Use the divergence theorem to convert the surface integration term into a volume integration term: CS CVB dA Bd    rr rContinuity EquationCSV dA 0( ) 0 using divergence theoremAlso, taking the CV to the limit of infinitesimally samll( ) 0 this is the continuity equation (mass conservation)In CCVCV CVdtd V dtVt              rrrr( ) ( ) ( )artesian coordiante: 0( ) ( ) ( )Special cases: steady state 0, ( ) 0,tIncompressible =constantV=0, V=0, 0u v wt x y zu v wVx y zu v wx y z                                 rr rPhysical Interpretationu vConsider two dimensional flow: 0, on a small fluid CV x yx y     u(x+ x)u(x)v(y+y)v(y)uSituation 1: 0, ( ) ( )More fluid leaving CV than enteringalong the x -direction, therefore, thereshould be more fluid entering thanleaving along the y-direction.vv(y) v(y y): 0yu x x u xx      uSituation 2: 0, ( ) ( ). More fluid entering CV than leavingalong the x -direction, therefore, there should be more fluid leaving thanventering along the y-direction. v(y) v(y y): 0yBothu x x u xx      u v situations should satisfy 0, mass conservationx y   Example (suction boundary layer control)A laminar boundary layer can be approximated as having a velocity profile u(x)=Uy/, where =cx1/2, c is a constant, U is the freestream velocity, and  is the boundary layer thickness. Determine the v(vertical component) of the velocity inside the boundary layer. (x)=cx1/2As the boundary layer grows downstream, the u-velocity is slowed downby the presence of viscous effect and the no-slip condition atthe solid surface. In order to satisfy the mass conservation equati1/ 2 3/ 2on, thev-velocity should be positive and removing the fluid away from the boundary layer.u v0,2Uy Uy vx y x cx c x y             Example (cont.)23/2 1/ 2UyIntegrate the v with respect to y: v4cx 4 4The velocity ratio: v/u y/4x increases away from the surface at a fixed xposition;it decreases further downstream at a fixed Uy y uycx x x           1/21/ 2y location.At the edge of the boundary layer y cx , /4It also decreases further downstream.cv ux  x( )x0 1 2 3 4 50123distanceboundary layer thicknessvratio x( )x0 1 2 3 4 50.10.20.30.4distancevelocity


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