The Laplace equation for potential function in fluid flow is formulated as:
Δ ϕ (x,y)=0, where Δ is the Laplacian operator. I want to know what does happen to the solution when ϕ and vector-field Ψ , have time-dependency?, it means how to solve Δ ϕ (x,y,t)=0. What is the name of the equation: Δ ϕ (x,y,t)=0.
I want to model time-dependent fluid velocity vector-field Ψ(x,y,t) and stream function to plot streamlines, and please also comment with regard to this.
Please also kindly suggest some articles and texts on this equation and also tell me as well what does happen when the problem is three-dimensional Δ ϕ (x,y,z,t)=0.
Article Three-dimensional Computational Fluid Dynamics Modeling of T...
https://www.sciencedirect.com/search?qs=How+to+model+and+simulate+time-dependent+two+or+three+dimensional+fluid+flow+using+stream+function&authors=&pub=&volume=&issue=&page=&origin=home&zone=qSearch
@Md. Washim Akram. Thank you Washim. I have wrongly prompted in my question for simulation, whereas indeed I am looking for analytic insight, not numerical simulation. Something like Eq. (15), in the article attached here.
Generating Exact Solution of Three Dimensional Coupled Unsteady Nonlinear Generalized Viscous Burgers’ Equations:
Article Generating Exact Solution of Three Dimensional Coupled Unste...
This is a good research on the topic, as I found it just right now.
You might find this paper, on 2-d unsteady viscid flow using streamfunctions, useful
Johnson, M.W., ‘Computation of Flow in a Vortex Shedding Flowmeter’, Jnl. of Flow Measurement and Instrumentation, Vol. 1, No. 4, July 1990, pp 183-190.
Using the velocity potential or the streamfunction limits the types of flow you can solve. The velocity potential can only be used for inviscid flows, but can be used for 2-d or 3-d flow. The streamfunction can be used for viscid steady or unsteady flow but is limited to 2-d flow.
@Mark Wyatt Johnson. Thank you.
The problem is that at your cited research:
Johnson, M.W., ‘Computation of Flow in a Vortex Shedding Flowmeter’, Jnl. of Flow Measurement and Instrumentation, Vol. 1, No. 4, July 1990, pp 183-190:
the nomenclature is a bit vague: The kinematic viscosity is v, and the velocity component in y direction is also v.
Moreover, the vorticity is the Laplacian of the stream-function with negative sign (Poisson's equation), not positive (as assumed in the paper), but these are interchangeable and not a mistake.
It generally seems a good research and I will look to study it for my own research. Regards.
The classical unsteady formulation for the potential function psi=psi(x,t) is coupled to the vorticity dynamics. That is you solve the system provided by the time-dependent equation for the vorticity field z=z(x,t)
dz/dt = convection + stretching + diffusion
Div Grad psi = z
with proper initial and boundary conditions.
Note that for the general 3D case you have 3 scalar parabolic equations for the vorticity components and 3 scalar elliptic equations for the components of the potential functions.
For this reason, the formulation in 3D is practically never used and is confined often only to 2D flows where rewrite in the x,y plane as
dzk/dt = convection + diffusion
Div Grad psik = zk
where now the component psik assumes the meaning of stream-function.
The stream-function/vorticity formulation is strictly connected to the Hodge decomposition theorem.
@professor Filippo Maria Denaro. Thanks. First I should remind you that, convection is a heat-transfer terminology, not fluid dynamics subject as I'm investigating. It seems you mean convective (not convection). I caught a glimpse over Hodge decomposition method, but it seems too abstract and theoretical.
Of course I mean convection of vorticity due to macroscopic transport of fluid.
The Hodge decomposition has several practical applications despite its theoretical illustration. The stream function - vorticity formulation is one of them. You can find details in one of my papers and its references.
@professor Filippo Maria Denaro. Thank you. Regards. Saeb.
There are also fundamental texbooks of fluid dynamics and CFD where you can find further details.
@professor Filippo Maria Denaro. Would you please name some of those text books?
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