AE/ME 339 Professor of Aerospace Engineering 12/21/01 topic7_ns_equations 1
Continuity equation Governing equation summary Non-conservation form D Dt. V 0.(2.29) Conservation form ( V ) 0...(2.33) t 12/21/01 topic7_ns_equations 2
Momentum equation Governing equation summary Non-conservation form Du p Dt x x y z xx yx zx f x...(2.50 a) Dv p Dt y x y z xy yy zy f y...(2.50 b) Dw p Dt z x y z xz yz zz f...(2.50 c) 12/21/01 topic7_ns_equations 3 z
Momentum equation Governing equation summary conservation form u p t x x y z xx yx zx uv fx...(2.56 a) v p t y x y z xy yy zy vv f y...(2.56 b) w p t z x y z xz yz zz wv fz...(2.56 c) 12/21/01 topic7_ns_equations 4
Energy equation Governing equation summary non-conservation form D 2 V. e q k T k T k T Dt 2 x x y y z z up vp wp u u yx u xx x y z x y z zx vxy v yy vzy w w yz w xz zz fv....(2.66) x y z x y z 12/21/01 topic7_ns_equations 5
Energy equation Governing equation summary conservation form 2 2 V V. T e e V q k t 2 2 x x up vp wp T T k k y y z z x y z u u yx u vxy v yy xx zx x y z x y vzy w w yz w xz zz fv....(2.81) z x y z 12/21/01 topic7_ns_equations 6
Exercise: Write the corresponding equations for inviscid flow. Observations: 1. Equations are coupled and non-linear 2. Conservation form contains divergence of some quantity on the LHS. This form is sometimes known as divergence form. 3. Normal and shear stress terms are functions of velocity gradient. 4. We have six unknowns and five equations (1 continuity + 3 momentum + 1 energy). For incompressible flow can be treated as a constant. For compressible flow, the equation of state can be used as an additional equation for the solution. 5. The set of equations with viscosity included, is known as the Navier-Stokes equations. 6. The set of inviscid flow equations is also known as the Euler equations. These naming conventions are not strictly followed by everyone. 12/21/01 topic7_ns_equations 7
Physical Boundary Conditions The above equations are very general. For example, they represent flow over an aircraft or flow in a hydraulic pump. To solve a specific problem much more information would be necessary. Some of them are listed below: 1. Boundary conditions (far field, solid boundary, etc) 2. Initial conditions (for unsteady problems) 3. Fluid medium (gas, liquid, non-newtonian fluid, etc.) BC specification depends on the type of flow we are interested in. e. g., velocity boundary condition at the surface No slip condition for viscous flow. All velocity components at the surface are zero. Zero normal velocity of inviscid flow. 12/21/01 topic7_ns_equations 8
Temperature BC at the wall. Temperature, Tw, heat flux, qw, etc. can be specified. Note T qw k n If q w is a known quantity, an expression for normal to the surface can be written in terms of known quantities. 12/21/01 topic7_ns_equations 9
Conservation form of the equations All equations can be expressed in the same generic form fluxes can be written as V uv vv wv ev 2 V ( e ) V 2 12/21/01 topic7_ns_equations 10
Conservation form contains divergence of these fluxes. U F G H J...(2.93) t x y z U u v w V e 2 2...(2.94) 12/21/01 topic7_ns_equations 11
F u 2 u pxx vuxy wu xz 2 V T e u puk u v w 2 x xx xy xz...(2.95) 12/21/01 topic7_ns_equations 12
v uv yx yy wv yz 2 V T e w pvk u v w 2 y 2 G v p yx yy yz...(2.96) H w uw zx vwzy 2 w pzz 2 V T e w pwk u v w 2 z zx zy zz...(2.97) 12/21/01 topic7_ns_equations 13
12/21/01 topic7_ns_equations 14 Computational Fluid Dynamics (AE/ME 339) 0...(2.98) x y z x y z f J f f uf vf wf q
In the above U is called the solution vector F, G, H are called flux vectors J is called the source term vector The problem is thus formulated as an unsteady problem. Steady state solutions can be obtained asymptotically. Once the flux variables are known from the solution, the primitive variables, u, v, w, p, e etc. can be obtained from the flux variables. Exercise write the vector form of the equations for inviscid flow (Euler equations). 12/21/01 topic7_ns_equations 15
Note that the following equations can be used to determine T e e p,...(2.112 a) RT R p ect v...(2.112 b) 1 1R 1 e 1 p 12/21/01 topic7_ns_equations 16
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