Numerical Simulations of Two-Layer Flow past Topography. Part II: Lee Vortices

2020 ◽  
Vol 77 (3) ◽  
pp. 965-980
Author(s):  
Richard Rotunno ◽  
George H. Bryan
Keyword(s):  
Shallow Water ◽  
Hydraulic Jump ◽  
Stokes Equations ◽  
Turbulent Energy ◽  
Navier Stokes ◽  
Constant Density ◽  
Vertical Vorticity ◽  
Lee Vortex ◽  
Spanwise Vorticity ◽  
Layer Flow

Abstract This study considers a two-layer fluid with constant density in each layer connected by a layer of continuously varying density for flows past topography in which hydraulic jumps with lee vortices are expected based on shallow-water theory. Numerical integrations of the Navier–Stokes equations at a Reynolds number high enough for a direct numerical simulation of turbulent flow allow an examination of the internal mechanics of the turbulent leeside hydraulic jump and how this mechanics is related to lee vortices. Analysis of the statistically steady state shows that the original source of lee-vortex vertical vorticity is through the leeside descent of baroclinically produced spanwise vorticity associated with the hydraulic jump. This spanwise vorticity is tilted to the vertical at the spanwise extremities of the leeside hydraulic jump. Turbulent energy dissipation in flow through the hydraulic jump allows this leeside vertical vorticity to diffuse and extend downstream. The present simulations also suggest a geometrical interpretation of lee-vortex potential-vorticity creation, a concept central to interpretations of lee vortices based on the shallow-water equations.

scholarly journals CAUCHY PROBLEM FOR VISCOUS SHALLOW WATER EQUATIONS WITH A TERM OF CAPILLARITY

2010 ◽  
Vol 20 (07) ◽  
pp. 1049-1087 ◽  
Author(s):  
BORIS HASPOT
Keyword(s):  
Shallow Water ◽  
Existence Of Solutions ◽  
Stokes Equations ◽  
Variable Viscosity ◽  
Water System ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Viscosity Coefficients ◽  
Global Existence Of Solutions ◽  
Dependent Viscosity

In this paper, we consider the compressible Navier–Stokes equation with density-dependent viscosity coefficients and a term of capillarity introduced formally by van der Waals in Ref. 51. This model includes at the same time the barotropic Navier–Stokes equations with variable viscosity coefficients, shallow-water system and the model introduced by Rohde in Ref. 46. We first study the well-posedness of the model in critical regularity spaces with respect to the scaling of the associated equations. In a functional setting as close as possible to the physical energy spaces, we prove global existence of solutions close to a stable equilibrium, and local in time existence of solutions with general initial data. Uniqueness is also obtained.

Two-dimensional global low-frequency oscillations in a separating boundary-layer flow

2008 ◽  
Vol 614 ◽  
pp. 315-327 ◽  
Author(s):  
UWE EHRENSTEIN ◽  
FRANÇOIS GALLAIRE
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Perturbation Analysis ◽  
Boundary Layer Flow ◽  
Stokes Equations ◽  
Low Frequency ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Optimal Perturbation ◽  
Layer Flow

A separated boundary-layer flow at the rear of a bump is considered. Two-dimensional equilibrium stationary states of the Navier–Stokes equations are determined using a nonlinear continuation procedure varying the bump height as well as the Reynolds number. A global instability analysis of the steady states is performed by computing two-dimensional temporal modes. The onset of instability is shown to be characterized by a family of modes with localized structures around the reattachment point becoming almost simultaneously unstable. The optimal perturbation analysis, by projecting the initial disturbance on the set of temporal eigenmodes, reveals that the non-normal modes are able to describe localized initial perturbations associated with the large transient energy growth. At larger time a global low-frequency oscillation is found, accompanied by a periodic regeneration of the flow perturbation inside the bubble, as the consequence of non-normal cancellation of modes. The initial condition provided by the optimal perturbation analysis is applied to Navier–Stokes time integration and is shown to trigger the nonlinear ‘flapping’ typical of separation bubbles. It is possible to follow the stationary equilibrium state on increasing the Reynolds number far beyond instability, ruling out for the present flow case the hypothesis of some authors that topological flow changes are responsible for the ‘flapping’.

scholarly journals Finite-Volume Solvers for a Multilayer Saint-Venant System

2007 ◽  
Vol 17 (3) ◽  
pp. 311-320 ◽  
Author(s):  
Emmanuel Audusse ◽  
Marie-Odile Bristeau
Keyword(s):  
Shallow Water ◽  
Numerical Investigation ◽  
Finite Volume ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Relevant Alternative ◽  
Navier Stokes Equations ◽  
Numerical Tests ◽  
Shallow Water Models ◽  
Water Models

Finite-Volume Solvers for a Multilayer Saint-Venant SystemWe consider the numerical investigation of two hyperbolic shallow water models. We focus on the treatment of the hyperbolic part. We first recall some efficient finite volume solvers for the classical Saint-Venant system. Then we study their extensions to a new multilayer Saint-Venant system. Finally, we use a kinetic solver to perform some numerical tests which prove that the 2D multilayer Saint-Venant system is a relevant alternative to 3D hydrostatic Navier-Stokes equations.

Models of non-Boussinesq lock-exchange flow

2011 ◽  
Vol 675 ◽  
pp. 1-26 ◽  
Author(s):  
R. ROTUNNO ◽  
J. B. KLEMP ◽  
G. H. BRYAN ◽  
D. J. MURAKI
Keyword(s):  
Free Boundary ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Water System ◽  
Analytic Solutions ◽  
Analytical Models ◽  
Navier Stokes ◽  
Exchange Flow

Nearly all analytical models of lock-exchange flow are based on the shallow-water approximation. Since the latter approximation fails at the leading edges of the mutually intruding fluids of lock-exchange flow, solutions to the shallow-water equations can be obtained only through the specification of front conditions. In the present paper, analytic solutions to the shallow-water equations for non-Boussinesq lock-exchange flow are given for front conditions deriving from free-boundary arguments. Analytic solutions are also derived for other proposed front conditions – conditions which appear to the shallow-water system as forced boundary conditions. Both solutions to the shallow-water equations are compared with the numerical solutions of the Navier–Stokes equations and a mixture of successes and failures is recorded. The apparent success of some aspects of the forced solutions of the shallow-water equations, together with the fact that in a real fluid the density interface is a free boundary, shows the need for an improved theory of lock-exchange flow taking into account non-hydrostatic effects for density interfaces intersecting rigid boundaries.

A Note on the Accuracy of Similar Solutions of the Equations for Laminar Boundary Layers on Curved Surfaces

1969 ◽  
Vol 73 (699) ◽  
pp. 226-228 ◽  
Author(s):  
B. S. Massey ◽  
B. R. Clayton
Keyword(s):  
Stokes Equations ◽  
Single Equation ◽  
Navier Stokes ◽  
Constant Density ◽  
Curved Surfaces ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Laminar Boundary Layers ◽  
Density Flow ◽  
Similar Solutions

Summary To describe steady, two-dimensional, constant-density flow in a laminar boundary layer on a curved surface, a single equation may be derived from the complete Navier-Stokes equations, with no approximations being necessary. The conditions under which similar solutions are attainable are discussed, and the validity of some previous calculations is upheld.

scholarly journals Lattice Boltzmann method for 2D flows in curvilinear coordinates

2012 ◽  
Vol 14 (3) ◽  
pp. 772-783 ◽  
Author(s):  
Ljubomir Budinski
Keyword(s):  
Shallow Water ◽  
Lattice Boltzmann ◽  
Shallow Water Equations ◽  
Stokes Equations ◽  
Complex Geometry ◽  
Standard Form ◽  
Curvilinear Coordinates ◽  
Navier Stokes ◽  
Equilibrium Distribution Function ◽  
Rectangular Lattice

In order to improve efficiency and accuracy, while maintaining an ease of modeling flows with the lattice Boltzmann approach in domains having complex geometry, a method for modeling equations of 2D flow in curvilinear coordinates has been developed. Both the transformed shallow water equations and the transformed 2D Navier-Stokes equations in the horizontal plane were synchronized with the equilibrium distribution function and the force term in the rectangular lattice. Since the solution of these equations takes place in the classical rectangular lattice environment, boundary conditions are modeled in the standard form of already existing simple methods (bounce-back), not requiring any additional functions. Owing to this and to the fact that the proposed method ensures a more accurate fitting of equations, even to domains of interest having complex geometry, the accuracy of solution is significantly increased, while the simplicity of the standard lattice Boltzmann approach is maintained. For the shallow water equations transformed in curvilinear coordinates, the proposed procedure is verified in three different hydraulic problems, all characterized by complex geometry.

A stationary circular hydraulic jump, the limits of its existence and its gasdynamic analogue

2008 ◽  
Vol 601 ◽  
pp. 189-198 ◽  
Author(s):  
ASLAN R. KASIMOV
Keyword(s):  
Surface Tension ◽  
Hydraulic Jump ◽  
Stokes Equations ◽  
Critical Value ◽  
Water Model ◽  
Navier Stokes ◽  
Jump Conditions ◽  
Shallow Water Model ◽  
Finite Radius ◽  
Navier Stokes Equations

We propose a theory of a steady circular hydraulic jump based on the shallow-water model obtained from the depth-averaged Navier–Stokes equations. The flow structure both upstream and downstream of the jump is determined by considering the flow over a plate of finite radius. The radius of the jump is found using the far-field conditions together with the jump conditions that include the effects of surface tension. We show that a steady circular hydraulic jump does not exist if the surface tension is above a certain critical value. The solution of the problem provides a basis for the hydrodynamic stability analysis of the hydraulic jump. An analogy between the hydraulic jump and a detonation wave is pointed out.

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