Vortex Transport Phenomena in Turbulent Fluid motion And role of Vortical Structures in Reynolds Stress Production and Distribution Yasmin Khakpour , Miad Yazdani Sharif University of Technology Tehran - Iran
Outline : � Momentum Transport Decomposition � Describing The Role of Acceleration Term in Reynolds Stress Distribution � Role of Vortex Structures near and far from the boundary into the acceleration term � Deviation Behaviors of Turbulence Viscosity Model � Vortex Transport Equation and the Role of stretching Term where Large Velocity gradients are significant � Comparisons between the results of Transport Equation and DNS results for vorticity fluxes in Separating Flows � Derivation of Vortex Method Equation in order to analyze the flow without any need to generate a mesh � Computing advection and stretching terms
Momentum Transport Decomposition - Introducing a fixed point P and an arbitrary point b, � We may write : � DX/Ds=U(X(s),s) X(s), denotes the particle displacement at time t and it can take any value � from point b to p . Dx(s)/Ds refers to Lagrangian velocity since X is differentiated with respect to time. Since U is denoted as the velocity at point p, it must be an Eulerian velocity , on � the other hand, X(s) is the local displacement and it cannot be a fixed distance, thus it must be considered Lagrangian . So U(X(s),s) is a compound value of Lagrangian and Eulerian velocity. � -U is randomized , from ٢ points of view: � First , because of the randomness of velocity field, � Second , because of the randomness of displacement field �
� By introducing τ b as the time taken for a particle moving from the arbitrary point b to the fixed point P, we have � At time t , the particle starting from point b , arrives at point P: U= Ū +u � At time t- τ b , the particle is at point b . U b = Ū b +u b
� By this assumption ,after rearranging the relations above and multiplying by v , we may write : __ ______ ________ __ uv = v( Ū b - Ū ) + v(U - U b ) + vu b � The first term is called displacement term, because it involves randomness of displacement and the second is named acceleration term ,as it denotes randomness of velocity field.
Reynolds Stress Decomposition in channel flow As it is evident from the figure ,the acceleration term near the boundary has a contract effect � since it reduces the Reynolds stress near the wall, while it has to be noted that the net behavior of Reynolds stress through the system is increasing.
Vortical structures near and far from the boundary Near the wall, where viscous effects are dominant, � because of vortex alignment in spanwise direction, the particles that their energy is dissipated, move through the wall and thus sweeping of the particles occurs. As we move through the wall, since v> ٠ and U-U b is negative ,the net contribution of acceleration term to Reynolds stress is negative.
� On the other hand, far from the boundary, where vortices are titled in streamwise direction because of mean field momentum, high energy containing vortices move away from the wall. In this case, where ejection becomes as the dominant effect, v> ٠ as before, but U-U b is positive and therefore the net correlation of these two is positive.
Reynolds Stress Decomposition in channel flow As it is evident from the figure ,the acceleration term near the boundary has a contract effect � since it reduces the Reynolds stress near the wall, while it has to be noted that the net behavior of Reynolds stress through the system is increasing.
Eddy viscosity model and its deficiencies � Writing Taylor series for U b around the fixed point: U b = U - L ٢ dU/dy + L ٢ ٢ (d ٢ U/dy ٢ ) Multiplying by v: � ___ ___ _________ ____________ (vU b - v U) = - vL ٢ d Ū /dy + vL ٢ ٢ (d ٢ Ū /dy ٢ ) � Comparing with previous relations for Reynolds stress , yields the fact that the eddy viscosity model is true if : ١ .Higher terms of Taylor series are neglected for displacement term, and ٢ .Acceleration term is thoroughly neglected . Actually it is not true particularly near the boundary, where the role of acceleration term is evidently different.
Vortex Transport Equation and Reynolds stress Distribution in separating flows � Taylor’s expression for Raynolds stress convection: � Introducing Taylor’s Equation: ∂ u i uj/ ∂ x j = ∂ K/ ∂ x i - ε ijk u j ω k And substituting in RANS equation: ∂ U i / ∂ t + U j ∂ U j / ∂ x j = - ∂ (P/ ρ + K) / ∂ x j + υΔ U j + ε ijk u j ω k � Assuming uni-directional flow (such as channel flows) we have: Ū = ( Ū , ٠ , ٠ ) Ω = ( ٠ , ٠ , Ω ٣ ) So above equation becomes: ٠ = - ١ / ρ ∂ P/ ∂ y + υ d ٢ U/dy ٢ + v ω ٣
� Taylor assumed that : v ω ٣ = vL ٢ d Ω ٣ /dy � Since the action of eddy viscousity is independent of the quality transported, we write coefficient above as it was appeared before.
� This assumption fails particularly where vortex structures play a major role in Reynolds stress distribution ,because in these cases the equation above cannot explain clearly the behavior of vortices. � To take vortical structures into account ,we rewrite vortex transport Equation as we’d done for momentum: _ _ v ω j = v( Ω j - Ω j b ) + v( Ω j - Ω j b ) + v ω j b (*) We call the first as Displacement term , and the second is called stretching term. Recalling vortex equation: ∂ Ω j/ ∂ t = Ω k (s) ∂ U j / ∂ x k + υΔΩ j (s) � Integrating Vortex Equation over the mixing time ,it follows : v( Ω j - Ω j b ) = v ∫ Ω k (s) ∂ U j / ∂ x k ds + v ∫ υΔΩ j (s) ds
� It shows that stretching term, by itself includes two different terms: v ∫ Ω k (s) ∂ U j / ∂ x k ds v ∫ υΔΩ j (s) ds � The first term is stretching because of convection and the second is diffusion due to viscousity effects. � Neglecting the second term,and substituting the Equation above into Equation for vorticity transport (Eq(*)) , we will have: _ _ b + v( Ω j - Ω j v ω j = v ω j b ) + v ∫ Ω k (s) ∂ U j / ∂ x k ds + v ∫ υΔΩ j (s) ds
� It is shown that Reynolds stress distribution is the result of stretching of vortices and their convection through the system. � The former is often neglected and this is why , viscosity models can not clearly predict the Reynolds stress distribution near the wall. � However ,since vortex Equation does not contain pressure gradient, it can be used for cases where we are encountered with inverse pressure gradient such as separating flows.
� The following figures show the vorticity fluxes for different vortex and momentum components: � The vorticity equation is solved for separating flows and then compared with DNS results for constant positive pressure gradient.
Comparisons of vorticity fluxes-u ω ٣ between DNS results and vorticity transport equations Inverse correlation of u and ω ٣ near the wall is evidently because of inverting flows near the boundary .
Comparisons of vorticity fluxes-w ω ١ between DNS results and vorticity transport equations Highly distributed vorticity fluxes near the boundary is clearly due to their accumulation near the wall , partcicularly for separating flows
Comparisons of vorticity fluxes-v ω٣ between DNS results and vorticity transport equations The contract effect of v ω٣ near the wall is presumably because of highly inverting flows there,that cause the invert direction of ω٣
Vortex methods � According to LES assumption, it is well accepted to consider just relatively few principal vortices in general flow configuration. � The idea that will be introduced, is based on Lagrangian transport of vortices and finally we’ll see that there will be no need to generate the mesh over the domain.
� Decomposing the velocity into two terms U ١ and U ٢ : U=U ١ +U ٢ Where, U ١ : solenoidal part of velocity Div U ١ = ٠ ... U ٢ : free rotational part of velocity Curl U ٢ = ٠
� vector potential, B, is defined as follows: Curl B = U ١ � We may introduce Ω as follows : Ω = - Δ B � Solving above equation for B and then for U ١ , yields to : U ١ = - ∫ R ( x-y ) * Ω / ׀ x-y ׀ ٣ dy
� U ٢ is irrotational part of velocity field and can be obtained by solving the Laplace equation as follows: U ٢ = Δφ ∂φ / ∂ n = n . (U-U ١ ) � By solving the Newmann problem, φ will be defined. � Solution of Newmann problem can be determined in the form of integral over the surface source distribution . � Since each of above equations can be solved analytically to obtain the velocity field, there will be no need to generate a grid over the domain.
Vortex elements � For the flow region away from the boundaries there are two main choices: � Vortex blubs which are the local volume of the fluid with vorticity of general form Ω (t) f h ( x - x ’ ) ; f h ( r ) = f( r/ h) . ١ /h ٣ � Vortex tubes which are special case of blubs , consisting a short, straight cylindrical volume with vorticities aligned in axial direction Ω = Σ N Ω i (t) f h ( x - x ’ ) : vorticity due to collection of number of tubes or blubs. Substituting into the velocity field equation, we can obtain velocity filed as follows: U( x , t) = Σ N Ω i (t) K( x - x ’ ) + U ٢
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