Two-dimensional flow of a viscous fluid in a channel with porous walls

1991 ◽  
Vol 227 ◽  
pp. 1-33 ◽  
Author(s):  
Stephen M. Cox
Keyword(s):  
Finite Time ◽  
Stokes Equations ◽  
Pitchfork Bifurcation ◽  
Time Dependent ◽  
High Rate ◽  
Navier Stokes ◽  
Recent Theory ◽  
Navier Stokes Equations ◽  
Two Dimensional Flow ◽  
Steady Solutions

We consider the flow of a viscous incompressible fluid in a parallel-walled channel, driven by steady uniform suction through the porous channel walls. A similarity transformation reduces the Navier-Stokes equations to a single partial differential equation (PDE) for the stream function, with two-point boundary conditions. We discuss the bifurcations of the steady solutions first, and show how a pitchfork bifurcation is unfolded when a symmetry of the problem is broken.Then we describe time-dependent solutions of the governing PDE, which we calculate numerically. We analyse these unsteady solutions when there is a high rate of suction through one wall, and the other wall is impermeable: there is a limit cycle composed of an explosive phase of inviscid growth, and a slow viscous decay. The inviscid phase ‘almost’ has a finite-time singularity. We discuss whether solutions of the governing PDE, which are exact solutions of the Navier-Stokes equations, may develop mathematical singularities in a finite time.When the rates of suction at the two walls are equal so that the problem is symmetrical, there is an abrupt transition to chaos, a ‘homoclinic explosion’, in the time-dependent solutions as the Reynolds number is increased. We unfold this transition by perturbing the symmetry, and compare direct numerical integrations of the governing PDE with a recent theory for ‘Lorenz-like’ dynamical systems. The chaos is found to be very sensitive to symmetry breaking.

scholarly journals Structure and stability of non-symmetric Burgers vortices

1998 ◽  
Vol 363 ◽  
pp. 199-228 ◽  
Author(s):  
AURELIUS PROCHAZKA ◽  
D. I. PULLIN
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Structure And Stability ◽  
Two Dimensional Flow ◽  
Steady Solutions ◽  
Pseudo Spectral Method ◽  
Burgers Vortices

We investigate, numerically and analytically, the structure and stability of steady and quasi-steady solutions of the Navier–Stokes equations corresponding to stretched vortices embedded in a uniform non-symmetric straining field, (αx, βy, γz), α+β+γ=0, one principal axis of extensional strain of which is aligned with the vorticity. These are known as non-symmetric Burgers vortices (Robinson & Saffman 1984). We consider vortex Reynolds numbers R=Γ/(2πv) where Γ is the vortex circulation and v the kinematic viscosity, in the range R=1−104, and a broad range of strain ratios λ=(β−α)/(β+α) including λ>1, and in some cases λ[Gt ]1. A pseudo-spectral method is used to obtain numerical solutions corresponding to steady and quasi-steady vortex states over our whole (R, λ) parameter space including λ where arguments proposed by Moffatt, Kida & Ohkitani (1994) demonstrate the non-existence of strictly steady solutions. When λ[Gt ]1, R[Gt ]1 and ε≡λ/R[Lt ]1, we find an accurate asymptotic form for the vorticity in a region 1<r/(2v/γ)1/2[les ]ε1/2, giving very good agreement with our numerical solutions. This suggests the existence of an extended region where the exponentially small vorticity is confined to a nearly cat's-eye-shaped region of the almost two-dimensional flow, and takes a constant value nearly equal to Γγ/(4πv)exp[−1/(2eε)] on bounding streamlines. This allows an estimate of the leakage rate of circulation to infinity as ∂Γ/∂t =(0.48475/4π)γε−1Γ exp (−1/2eε) with corresponding exponentially slow decay of the vortex when λ>1. An iterative technique based on the power method is used to estimate the largest eigenvalues for the non-symmetric case λ>0. Stability is found for 0[les ]λ[les ]1, and a neutrally convective mode of instability is found and analysed for λ>1. Our general conclusion is that the generalized non-symmetric Burgers vortex is unconditionally stable to two-dimensional disturbances for all R, 0[les ]λ[les ]1, and that when λ>1, the vortex will decay only through exponentially slow leakage of vorticity, indicating extreme robustness in this case.

Singularities

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Stokes Equations ◽  
Negative Answer ◽  
Three Dimensions ◽  
Picard Iteration ◽  
Navier Stokes ◽  
Scale Invariant ◽  
Navier Stokes Equations ◽  
L2 Norm ◽  
Two Dimensional Flow ◽  
Physical Consequences

The initial value problem of the incompressible Navier–Stokes equations is explained. Leray’s classic study of it (using Picard iteration) is simplified and described in the language of physics. The ideas of Lebesgue and Sobolev norms are explained. The L2 norm being the energy, cannot increase. This gives sufficient control to establish existence, regularity and uniqueness in two-dimensional flow. The L3 norm is not guaranteed to decrease, so this strategy fails in three dimensions. Leray’s proof of regularity for a finite time is outlined. His attempts to construct a scale-invariant singular solution, and modern work showing this is impossible, are then explained. The physical consequences of a negative answer to the regularity of Navier–Stokes solutions are explained. This chapter is meant as an introduction, for physicists, to a difficult field of analysis.

scholarly journals A Code for Simulating Heat Transfer in Turbulent Channel Flow

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 756
Author(s):  
Federico Lluesma-Rodríguez ◽  
Francisco Álcantara-Ávila ◽  
María Jezabel Pérez-Quiles ◽  
Sergio Hoyas
Keyword(s):  
Heat Transfer ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Turbulent Channel Flow ◽  
Passive Scalar ◽  
Direct Numerical Simulations ◽  
Time Dependent ◽  
Navier Stokes ◽  
Thermal Flow ◽  
Navier Stokes Equations

One numerical method was designed to solve the time-dependent, three-dimensional, incompressible Navier–Stokes equations in turbulent thermal channel flows. Its originality lies in the use of several well-known methods to discretize the problem and its parallel nature. Vorticy-Laplacian of velocity formulation has been used, so pressure has been removed from the system. Heat is modeled as a passive scalar. Any other quantity modeled as passive scalar can be very easily studied, including several of them at the same time. These methods have been successfully used for extensive direct numerical simulations of passive thermal flow for several boundary conditions.

Investigation of the stability of boundary layers by a finite-difference model of the Navier—Stokes equations

1976 ◽  
Vol 78 (2) ◽  
pp. 355-383 ◽  
Author(s):  
H. Fasel
Keyword(s):  
Finite Difference ◽  
Stability Theory ◽  
Stokes Equations ◽  
Time Dependent ◽  
Linear Stability Theory ◽  
Navier Stokes ◽  
Experimental Measurements ◽  
Navier Stokes Equations ◽  
The Stability ◽  
Small Periodic

The stability of incompressible boundary-layer flows on a semi-infinite flat plate and the growth of disturbances in such flows are investigated by numerical integration of the complete Navier–;Stokes equations for laminar two-dimensional flows. Forced time-dependent disturbances are introduced into the flow field and the reaction of the flow to such disturbances is studied by directly solving the Navier–Stokes equations using a finite-difference method. An implicit finitedifference scheme was developed for the calculation of the extremely unsteady flow fields which arose from the forced time-dependent disturbances. The problem of the numerical stability of the method called for special attention in order to avoid possible distortions of the results caused by the interaction of unstable numerical oscillations with physically meaningful perturbations. A demonstration of the suitability of the numerical method for the investigation of stability and the initial growth of disturbances is presented for small periodic perturbations. For this particular case the numerical results can be compared with linear stability theory and experimental measurements. In this paper a number of numerical calculations for small periodic disturbances are discussed in detail. The results are generally in fairly close agreement with linear stability theory or experimental measurements.

An Exact Solution of the Unsteady Navier-Stokes Equations Resulting From a Stretching Surface

1994 ◽  
Vol 61 (3) ◽  
pp. 629-633 ◽  
Author(s):  
S. H. Smith
Keyword(s):  
Exact Solution ◽  
Stokes Equations ◽  
Limited Range ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Stretching Surface ◽  
Navier Stokes Equations ◽  
Short Period ◽  
Two Dimensional Flow ◽  
Surrounding Fluid

When a stretching surface is moved quickly, for a short period of time, a pulse is transmitted to the surrounding fluid. Here we describe an exact solution in terms of a similarity variable for the Navier-Stokes equations which represents the effect of this pulse for two-dimensional flow. The unusual feature is that this solution is only valid for a limited range of the Reynolds number; outside this domain unbounded velocities result.

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