Large Kinetic Reynolds Number Research Articles

Exact Navier-Stokes Solution for the Pulsatory Viscous Channel Flow with Arbitrary Pressure Gradient

In this article, an exact solution of the Navier–Stokes equations is presented for the motion of an incompressible viscous fluidinachannelwitharbitrarypressuredistribution.Ageneralizationispursuedbyexpressingthepressure signal in terms of Fourier coefficients. The flow is characterized by two principal parameters: the pulsation parameter based on the periodic pressure gradient, and the kinetic Reynolds number based on the pulsation frequency. By way of verification, it is shown that for large kinetic Reynolds numbers, the Poiseuille flow may be recovered. Conversely, the purely oscillatory solution by Rott is regained for large pulsation parameters. For sinusoidalpulsations,the flowrateisdeterminedandcomparedwiththesteady flowanalogobtainedunderthesame pressure gradient. The amount of flow amplification or attenuation is determined as a function of frequency and pulsation parameters. For large frequencies, the maximum flow rate is evaluated and related to the pulsation parameter. By characterizing the velocity, vorticity, and shear stress distributions, cases of flow reversal are identified. Conditions leading to flow reversal are delineated for small and large kinetic Reynolds numbers in both planar and axisymmetric chambers. For an appreciable pulsation rate, we find that the flow reverses when the pulsation parameter is increased to the point of exceeding the Stokes number. By evaluating the skin friction coefficient and its limiting value, design criteria for minimizing viscous losses are realized. The optimal frequency that maximizes the flow rate during pulsing is also determined. Finally, to elucidate the effect of curvature, comparisons between planar and axisymmetric flow results are undertaken. The family of exact solutions presented here can thus be useful in verifying and validating computational models of complex unsteady motions in both propulsiveandnonpropulsive applications. Theycan alsobeused toguide the designoffuel injectors andcontrolled experiments aimed at investigating the transitional behavior of periodic flows.

Anisotropic form of third-order moments and relationship to the cascade rate in axisymmetric magnetohydrodynamic turbulence

Laws governing the behavior of statistical third-order moments in the inertial range are among the few rigorous results in the theory of statistically homogeneous incompressible magnetohydrodynamic turbulence. These fundamental laws apply to both isotropic and anisotropic turbulence. Assuming that the turbulence is stationary in time and statistically axisymmetric under proper rotations about the direction of the mean magnetic field, it is shown that the general mathematical form of the tensor quantities appearing in these laws is constrained by symmetry to have a particular form. Using these forms, the general solutions of the law for the vector and pseudovector third-order moments F and FC are obtained in the limit of large kinetic Reynolds number and large magnetic Reynolds number. The physical meaning of the different terms in F and FC are investigated and a method for obtaining the cascade rates of energy ϵ and cross-helicity ϵC from experimental data is described. The results show that the measurement of the cascade rates goes hand-in-hand with the measurement of the spatial anisotropy of the third-order moments F and FC. The theory developed here can be applied to measure the turbulent cascade rates of energy and cross-helicity in laboratory plasma experiments, numerical simulations, and the solar wind.

Simulation of induction at low magnetic Prandtl number.

We consider the induction of a magnetic field in flows of an electrically conducting fluid at low magnetic Prandtl number and large kinetic Reynolds number. Using the separation between the magnetic and kinetic diffusive length scales, we propose a new numerical approach. The coupled magnetic and fluid equations are solved using a mixed scheme, where the magnetic field fluctuations are fully resolved and the velocity fluctuations at small scale are modeled using a large eddy simulation (LES) scheme. We study the response of a forced Taylor-Green flow to an externally applied field: topology of the mean induction and time fluctuations at fixed locations. The results are in remarkable agreement with existing experimental data; a global 1/f behavior at long times is also evidenced.

The Dependence of Dynamo α‐Effect on Reynolds Numbers, Magnetic Prandtl Number, and the Statistics of Magnetohydrodynamic Turbulence

We generalize the derivation of the dynamo coefficient α of Field et al. to include the following two aspects: first, the decorrelation times of velocity field and magnetic field are different; second, the magnetic Prandtl number can be arbitrary. We find that the contributions of velocity field and magnetic field to the α-effect are not equal, but affected by their different statistical properties. In the limit of large kinetic Reynolds number and large magnetic Reynolds number, α-coefficient may not be small if the decorrelation times of velocity field and magnetic field are shorter than the eddy turnover time of the MHD turbulence. We also show that under certain circumstances, for example if the kinetic helicity and current helicity are comparable, α depends insensitively on magnetic Prandtl number, while if either the kinetic helicity or the current helicity is dominated by the other one, a different magnetic Prandtl number will significantly change the dynamo α-effect.

Vorticity Dynamics in Isobarically Closed Porous Channels Part I: Standard Perturbations

When acoustic pressure oscillations are induced in a porous channel of the closed ‐open type, the linearized Navier‐Stokes equations can be solved analytically to obtain an accurate description of the temporal e owe eld corresponding to laminar conditions. The channel considered herehas a rectangularcross section, has two equally permeablewalls,andisopenatthedownstreamend.Thecurrentmethodologyparallelstheclosed ‐closedboundary analysis carried out previously for a channel with both ends closed. Limiting our scope to laminar conditions, we apply standard perturbation tools to present a closed-form solution that becomes asymptotically exact for large kinetic Reynolds numbers. Verie cations are materialized by way of numerical simulations and quantie cation of the maximum absolute error entailed in the e nal formulations. Note that the error is found to exhibit a clear asymptotic behavior. Furthermore, the analytical formulation reveals vortical structures and explains the direct ine uence of acoustic pressure oscillations on the rotational waves generated in the closed ‐open cone guration. Finally, the explicit roles of variable injection, viscosity, and oscillation frequency are explained.