New Representation of Navier-Stokes Equations Governing Self-Similar Homogeneous Turbulence

1983 ◽  
Vol 51 (8) ◽  
pp. 617-620 ◽  
Author(s):  
C. Foias ◽  
O. P. Manley ◽  
R. Temam
Keyword(s):  
Stokes Equations ◽  
Homogeneous Turbulence ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Self Similar

GLOBAL EXISTENCE TO THE NAVIER–STOKES EQUATIONS THROUGH A SELF-SIMILAR-LIKE UPPER SOLUTION

2012 ◽  
Vol 14 (05) ◽  
pp. 1250031
Author(s):  
GUY BERNARD
Keyword(s):  
Global Existence ◽  
Heat Equation ◽  
Initial Velocity ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Dimensional Euclidean Space ◽  
Time Interval ◽  
Navier Stokes Equations ◽  
Upper Solution ◽  
Self Similar

A global existence result is presented for the Navier–Stokes equations filling out all of three-dimensional Euclidean space ℝ3. The initial velocity is required to have a bell-like form. The method of proof is based on symmetry transformations of the Navier–Stokes equations and a specific upper solution to the heat equation in ℝ3× [0, 1]. This upper solution has a self-similar-like form and models the diffusion process of the heat equation. By a symmetry transformation, the problem is transformed into an equivalent one having a very small initial velocity. Using the upper solution, the equivalent problem is then solved in the time interval [0, 1]. This local solution is then extended to the time interval [0, ∞) by an iterative process. At each step, the problem is extended further in time in an interval of time whose length is greater than one, thus producing the global solution. Each extension is transformed, by an appropriate change of variables, into the first local problem in the time interval [0, 1]. These transformations exploit the diffusive and self-similar-like nature of the upper solution.

Natural convection flow far from a horizontal plate

1999 ◽  
Vol 387 ◽  
pp. 227-254 ◽  
Author(s):  
VALOD NOSHADI ◽  
WILHELM SCHNEIDER
Keyword(s):  
Boundary Layer ◽  
Prandtl Number ◽  
Stokes Equations ◽  
Axisymmetric Flow ◽  
The Self ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Boundary Layer Equations ◽  
Self Similar ◽  
Similar Solutions

Plane and axisymmetric (radial), horizontal laminar jet flows, produced by natural convection on a horizontal finite plate acting as a heat dipole, are considered at large distances from the plate. It is shown that physically acceptable self-similar solutions of the boundary-layer equations, which include buoyancy effects, exist in certain Prandtl-number regimes, i.e. 0.5<Pr[les ]1.470588 for plane, and Pr>1 for axisymmetric flow. In the plane flow case, the eigenvalues of the self-similar solutions are independent of the Prandtl number and can be determined from a momentum balance, whereas in the axisymmetric case the eigenvalues depend on the Prandtl number and are to be determined as part of the solution of the eigenvalue problem. For Prandtl numbers equal to, or smaller than, the lower limiting values of 0.5 and 1 for plane and axisymmetric flow, respectively, the far flow field is a non-buoyant jet, for which self-similar solutions of the boundary-layer equations are also provided. Furthermore it is shown that self-similar solutions of the full Navier–Stokes equations for axisymmetric flow, with the velocity varying as 1/r, exist for arbitrary values of the Prandtl number.Comparisons with finite-element solutions of the full Navier–Stokes equations show that the self-similar boundary-layer solutions are asymptotically approached as the plate Grashof number tends to infinity, whereas the self-similar solution to the full Navier–Stokes equations is applicable, for a given value of the Prandtl number, only to one particular, finite value of the Grashof number.In the Appendices second-order boundary-layer solutions are given, and uniformly valid composite expansions are constructed; asymptotic expansions for large values of the lateral coordinate are performed to study the decay of the self-similar boundary-layer flows; and the stability of the jets is investigated using transient numerical solutions of the Navier–Stokes equations.

scholarly journals Forward self-similar solutions of the fractional Navier-Stokes equations

2019 ◽  
Vol 352 ◽  
pp. 981-1043 ◽  
Author(s):  
Baishun Lai ◽  
Changxing Miao ◽  
Xiaoxin Zheng
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Self Similar ◽  
Similar Solutions

Particle dynamics and mixing in a viscously decaying shear layer

1991 ◽  
Vol 227 ◽  
pp. 211-244 ◽  
Author(s):  
E. Meiburg ◽  
P. K. Newton
Keyword(s):  
Exact Solution ◽  
Blow Up ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Core Size ◽  
Mixing Processes ◽  
Navier Stokes Equations ◽  
Interface Generation ◽  
Self Similar ◽  
Detailed Picture

We study the mixing of fluid in a viscously decaying row of point vortices. To this end, we employ a simplified model based on Stuart's (1967) one-parameter family of solutions to the steady Euler equations. Our approach relates the free parameter to a vortex core size, which grows in time according to the exact solution of the Navier-Stokes equations for an isolated vortex. In this way, we approach an exact solution for small values of t/Re. We investigate how the growing core size leads to a shrinking of the cat's eye and hence to fluid leaking out of the trapped region into the free streams. In particular, we observe that particles initially located close to each other in neighbouring intervals along the streamwise direction escape from the cat's eye near opposite ends. The size of these intervals scales with the inverse square root of the Reynolds number. We furthermore examine the particle escape times and observe a self-similar blow-up for the particles near the border between two adjacent intervals. This can be explained on the basis of a simple stagnation-point flow. An investigation of interface generation shows that viscosity leads to an additional factor proportional to time in the growth rates. Numerical simulations confirm the above results and give a detailed picture of the underlying mixing processes.

Similarity solutions for viscous vortex cores

1992 ◽  
Vol 238 ◽  
pp. 487-507 ◽  
Author(s):  
Ernst W. Mayer ◽  
Kenneth G. Powell
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Axial Flow ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
The Core ◽  
Axial Pressure Gradient ◽  
Self Similar ◽  
Similar Solutions

Results are presented for a class of self-similar solutions of the steady, axisymmetric Navier–Stokes equations, representing the flows in slender (quasi-cylindrical) vortices. Effects of vortex strength, axial gradients and compressibility are studied. The presence of viscosity is shown to couple the parameters describing the core growth rate and the external flow field, and numerical solutions show that the presence of an axial pressure gradient has a strong effect on the axial flow in the core. For the viscous compressible vortex, near-zero densities and pressures and low temperatures are seen on the vortex axis as the strength of the vortex increases. Compressibility is also shown to have a significant influence upon the distribution of vorticity in the vortex core.

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