Governing Equations for Turbulent Flow
(i) Boundary Layer on a Flat Plate ρu x Re x = = Reynolds Number µ Re Re x =5(10) 5 Re x =10 6 x =0 u/ U = 0.99 層流區域 過渡區域 紊流區域 Thickness of boundary layer
The Origin of Turbulence Turbulence is believed to be induced by small disturbance in flow. In laminar flow region, the disturbance is restrained by the flow, thus it does not affect the flow motion. In turbulent flow region, the disturbance is amplified by the flow motion to form eddies. For laminar flow, as the derived continuity, momentum and energy equations are used for predicting flow velocity and temperature distributions. The result appears to match well with experimental data. Nevertheless, for turbulent flow, this match usually appears to be poor. Reynolds number can be used for indicating the occurrence of turbulent flow.
Fluctuation of velocity and temperature in turbulent flow region u u at a certain location t 1 period t (time) Define: ' ' ' ' u = u + u (t); v = v + v (t); w = w + w(t); P = P + P (t) ' T = T + T (t) 1 1 1 w t t t t t t 1 1 1 u u dt, v v dt, dt etc. 0 0 0 1 1 1 w
(ii) Time-Averaged Continuity Equation u v w Continuity: + + = 0 (incompressible flow) x z ' ' ' u = u + u (t); v = v + v (t); w = w + w(t); substituting into the above equation, u u' v v' w w' + + + + + = x x z z 0 Integrating over period 0 t u u' v v w w' ' + + + + + = x x z z = 0 = 0 = 0 u v w + + = 0 x z 1 0
(iii) Time-Averaged x-direction Momentum Equation x-direction momentum equation (neglecting body force): u u u u 1 P e u u u + u + v + w = + ν ( + + ) t x z ρ x x z Multiplying continuity equation by u: u v w u ( + + ) = 0 adding to the above equation x z u u ( uv) ( uw) 1 P e u u u + + + = + ν ( + + ) t x z ρ x x z
averaging over period 0 t 1 u u ( uv) ( uw) 1 P e u u u + + + = + ν ( + + ) t x z ρ x x z = 0 u ( uv) ( uw) 1 P e u u u + + = + ν ( + + ) x z ρ x x z ' ' ' u = u + u (t); v = v + v (t); w = w + w(t); substituting these three expressions into the above equation, ( ) ( ') ( v) ( ' ') ( ) ( ' ') 1 [ u u u uv u w uw Pe u u u + ] + [ + ] + [ + ] = + ν ( + + ) x x z z ρ x x z
where ( ) ( v) ( ) u u u v u u w w u = u ; = u + v ; = u + w x x z z z Substituting back, u ( u' ) v u ( uv ' ') w u ( uw ' ') [ u + ] + [ u + v + ] + [ u + w + ] x x z z z 1 P e u u u = + ν ( + + ) (1) ρ x x z Multiplying the time-averaged continuity equation by u, u v w u ( + + ) = 0 x z Substracting equation (1) with this equation, it yields u ( u' ) u ( uv ' ') u ( uw ' ') 1 P e u u u [ u + ] + [v + ] + [ w + ] = + ν ( + + ) x x z z ρ x x z
After arrangement, 1 ( ') ( ' ') ( ' ') u u u P u uv uw u + v + w + = ν ( u) x z ρ x x z where u u u x z u + + Additional terms Similarly, y and z-directional momentum equations can be derived as well! In tensor notation, i-direction time-averaged momentum equation is u j ( uv ' ' i j) u 1 i P ui + = ν ( ) x ρ x x x x j j j j
For two-dimensional turbulent boundary layer, the x-momentum equation is, u u P u uv ρ 1 ( ' ') u + v + ν x y x y y u u ( u' ) ( uv ' ') where and x x (velocity fluctuation in the y direction is larger than that in the x direction) where u u 1 P 1 u + x y ρ x ρ y u µ = ρuv ε M v + = ( ' ') molecular shear stress ; ' ' e eddy momentum diffusivity = u µ ρuv Negative value u ddy shear stress ρε M ( ) u v u y [Note: velocity fluctuations ( ', ') are assumed to be induced by /.]
Hence, u u 1 P 1 u u u + v + = ( µ + ρεm ) x y ρ x ρ 1 u = [ ρν ( + εm ) ] ρ u = [( ν + εm ) ] where u ρν ( + εm ) = apparent shear stress
(iv) k ε Model for Turbulent Flow ε M is not a constant. For solving the momentum equation, its value must be determined. k ε Model Involves (i) Turbulence kinetic energy (k) equation (ii) Dissipation energy (ε) equation The above two equations plus the momentum equation can be used to solve the ε M value and velocity distribution.
Analog to molecular kinetic theory for gas: ε M = Ck u 1/ L where k = turbulence energy ; L = length scale ; C = empirical coefficient 3/ k Dissipation rate = ε = C D( ) L Sustituting back to eliminate L, it yields ε M = ( ) C u 1 D u where C = drag coefficient = 1 Dk ε M k u k equation: = ( ) + εm ( ) ε Dt σ k Dε εm ε u ε ε ε equation: = ( ) + c1ε M ( ) c ( ) Dt σ k k The following values are suggested for : C = 0.09, c = 1.44, c = 1.9, σ = 1, σ = 1.3 u ε k k ε ε
(v) Time-Averaged Energy Equation Energy equation (neglect dissipation term): T T T T T T T ρcp ( + u + v + w ) = k( + + ) t x z x z ' ' ' where u = u + u (t) ; v = v + v (t) ; w = w + w(t) ' ' P = P + P (t) ; T = T + T (t) Multiplying continuity equation by T: u v w T ( + + ) = 0 adding to the above equation x z T ( ut) ( vt) ( wt) T T T + + + = α( + + ) t x z x z
Time averaged from 0 t, 1 T ( ut) ( vt) ( wt) T T T + + + = α( + + ) t x z x z ( ut) ( vt) ( wt) T T T After arrangement, + + = α ( + + ) x z x z T u T v T w [ u + T ] + [ v + T ] + [ w + T ] x x z z ' ' ' ' ' ' ( ut ) ( vt ) ( wt ) T T T + + + = α( + + ) x z x z For two-dimensional flow (also using continuity equation), it yields : ' ' T T ( vt ) T u + v + = α x ' ' ' ' ( ) ( ) [Note: In the boundary layer, T T ut vt ; ] x x
For two-dimensional flow in the boundary layer: ' ' T T T ( vt ) u + v = α x 1 T ' ' T = [ k ρcpvt ] = [( α + εh) ] ρcp where T k = molecular heat flux (without considering direction) ' ' T ρcvt p = eddy heat flux ρcpεh ε eddy thermal diffusivity ; α = molecular thermal diffusivity H u T T ε M T + v = [( α + ) ] x Pr t where Pr t turbulent Prandtl number eddy momentum diffusivity ε = eddy thermal diffusivity ε M H (empirally determined)