Numerical Integration of the Navier-Stokes Equations for Steady Two-Dimensional Flow

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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Two-Dimensional Computational Model for Wave Rotor Flow Dynamics

Journal of Engineering for Gas Turbines and Power ◽

10.1115/1.2817086 ◽

1997 ◽

Vol 119 (4) ◽

pp. 978-985 ◽

Cited By ~ 26

Author(s):

G. E. Welch

Keyword(s):

Stokes Equations ◽

Flow Simulation ◽

Navier Stokes ◽

Two Dimensional ◽

Wave Rotor ◽

Navier Stokes Equations ◽

Flow Features ◽

Rotor Experiment ◽

Two Dimensional Flow ◽

Rotor Flow

A two-dimensional (θ, z) Navier–Stokes solver for multiport wave rotor flow simulation is described. The finite-volume forms of the unsteady thin-layer Navier–Stokes equations are integrated in time on multiblock grids that represent the stationary inlet and outlet ports and the moving rotor passages of the wave rotor. Computed results are compared with three-port wave rotor experimental data. The model is applied to predict the performance of a planned four-port wave rotor experiment. Two-dimensional flow features that reduce machine performance and influence rotor blade and duct wall thermal loads are identified.

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Turbulent 2.5-dimensional dynamos

Journal of Fluid Mechanics ◽

10.1017/jfm.2016.366 ◽

2016 ◽

Vol 799 ◽

pp. 246-264 ◽

Cited By ~ 4

Author(s):

K. Seshasayanan ◽

A. Alexakis

Keyword(s):

Turbulent Flows ◽

Large Scale ◽

Stokes Equations ◽

Three Dimensional ◽

Reynolds Numbers ◽

Navier Stokes ◽

Two Dimensional ◽

Navier Stokes Equations ◽

Wide Range ◽

Two Dimensional Flow

We study the linear stage of the dynamo instability of a turbulent two-dimensional flow with three components $(u(x,y,t),v(x,y,t),w(x,y,t))$ that is sometimes referred to as a 2.5-dimensional (2.5-D) flow. The flow evolves based on the two-dimensional Navier–Stokes equations in the presence of a large-scale drag force that leads to the steady state of a turbulent inverse cascade. These flows provide an approximation to very fast rotating flows often observed in nature. The low dimensionality of the system allows for the realization of a large number of numerical simulations and thus the investigation of a wide range of fluid Reynolds numbers $Re$, magnetic Reynolds numbers $Rm$ and forcing length scales. This allows for the examination of dynamo properties at different limits that cannot be achieved with three-dimensional simulations. We examine dynamos for both large and small magnetic Prandtl-number turbulent flows $Pm=Rm/Re$, close to and away from the dynamo onset, as well as dynamos in the presence of scale separation. In particular, we determine the properties of the dynamo onset as a function of $Re$ and the asymptotic behaviour in the large $Rm$ limit. We are thus able to give a complete description of the dynamo properties of these turbulent 2.5-D flows.

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Structure and stability of non-symmetric Burgers vortices

Journal of Fluid Mechanics ◽

10.1017/s0022112098008866 ◽

1998 ◽

Vol 363 ◽

pp. 199-228 ◽

Cited By ~ 15

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.

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