Dear all
Is there any analytical formulation based on Poiseuille equations
for velocity distribution of fluid flow through the curved duct with rectangular cross section?
Best Regards
Steady laminar flow through a pipe of uniform cross-section is called Poiseuille flow. In this case the velocity profile is a function of the radial coordinate only. The pressure gradient that drives the flow is a constant , i.e., the fluid pressure decreases linearly in the flow direction. A similar flow occurs in a channel with a constant cross-section. In this case it is called plane Poiseuille flow. In both cases the flow becomes unstable (i.e., transitions to a turbulent flow structure at some critical Reynolds number (Re ~ 2000 for pipe flow, Re~ 5772 for a channel). Laminar low in a curved pipe or channel is often referred to as Dean Flow. Because of the centripetal force that is generated due to the curvature of the pipe/channel, an adverse pressure gradient is generated which drives a secondary flow which is superimposed on the primary flow. Consequently the flow is no longer unidirectional, i.e, the flow field now depends on the coordinate in the flow direction. The secondary flow due to curvature effects appears as two counter-rotating eddies, call Dean vortices. These vortices appear at some critical Dean number.
So the answer to your question: there is no Poiseuille flow in a curved duct, as the flow now depends on two dimensionless parameters the Re number and a Dean number: De=Re Sqrt(D/2R_c), where Re is the usual Reynolds number and R_c is the radius of curvature of the pipe. For small De, the equations can be solved as a powers series in De, the leading term is the Hagen-Poiseuille formula! Thus for small De the flow is a perturbation from the Hagen Poisuille formula, and beyond the critical Dean number, Dean vortices, due to curvature effects, appear. Hope this helps.
In a design rectangular duct u must have the velocity and these velocities founded in any duct design book
Analytical solution for straight duct with rectangular section is available but if the duct is curved arbitrarily you cannot get that.
Steady laminar flow through a pipe of uniform cross-section is called Poiseuille flow. In this case the velocity profile is a function of the radial coordinate only. The pressure gradient that drives the flow is a constant , i.e., the fluid pressure decreases linearly in the flow direction. A similar flow occurs in a channel with a constant cross-section. In this case it is called plane Poiseuille flow. In both cases the flow becomes unstable (i.e., transitions to a turbulent flow structure at some critical Reynolds number (Re ~ 2000 for pipe flow, Re~ 5772 for a channel). Laminar low in a curved pipe or channel is often referred to as Dean Flow. Because of the centripetal force that is generated due to the curvature of the pipe/channel, an adverse pressure gradient is generated which drives a secondary flow which is superimposed on the primary flow. Consequently the flow is no longer unidirectional, i.e, the flow field now depends on the coordinate in the flow direction. The secondary flow due to curvature effects appears as two counter-rotating eddies, call Dean vortices. These vortices appear at some critical Dean number.
So the answer to your question: there is no Poiseuille flow in a curved duct, as the flow now depends on two dimensionless parameters the Re number and a Dean number: De=Re Sqrt(D/2R_c), where Re is the usual Reynolds number and R_c is the radius of curvature of the pipe. For small De, the equations can be solved as a powers series in De, the leading term is the Hagen-Poiseuille formula! Thus for small De the flow is a perturbation from the Hagen Poisuille formula, and beyond the critical Dean number, Dean vortices, due to curvature effects, appear. Hope this helps.
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