scholarly journals Derivation of Prandtl boundary layer equations for the incompressible Navier–Stokes equations in a curved domain

2014 ◽  
Vol 34 ◽  
pp. 81-85 ◽  
Author(s):  
Cheng-Jie Liu ◽  
Ya-Guang Wang
Keyword(s):  
Boundary Layer ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Boundary Layer Equations ◽  
Prandtl Boundary Layer

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 The numerical solution of the asymptotic equations of trailing edge flow

1974 ◽  
Vol 340 (1620) ◽  
pp. 91-111 ◽  
Keyword(s):  
Boundary Layer ◽  
Outer Layer ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Trailing Edge ◽  
Navier Stokes Equations ◽  
Boundary Layer Equations ◽  
Asymptotic Equations ◽  
Displacement Thickness

According to Stewartson (1969, 1974) and to Messiter (1970), the flow near the trailing edge of a flat plate has a limit structure for Reynolds number Re →∞ consisting of three layers over a distance O (Re -3/8 ) from the trailing edge: the inner layer of thickness O ( Re -5/8 ) in which the usual boundary layer equations apply; an intermediate layer of thickness O ( Re -1/2 ) in which simplified inviscid equations hold, and the outer layer of thickness O ( Re -3/8 ) in which the full inviscid equations hold. These asymptotic equations have been solved numerically by means of a Cauchy-integral algorithm for the outer layer and a modified Crank-Nicholson boundary layer program for the displacement-thickness interaction between the layers. Results of the computation compare well with experimental data of Janour and with numerical solutions of the Navier-Stokes equations by Dennis & Chang (1969) and Dennis & Dunwoody (1966).

scholarly journals Spectral stability of Prandtl boundary layers: An overview

Analysis ◽  
2015 ◽  
Vol 35 (4) ◽  
Author(s):  
Emmanuel Grenier ◽  
Yan Guo ◽  
Toan T. Nguyen
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Asymptotic Expansions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Stability Of Shear Flows ◽  
The Stability ◽  
Prandtl Boundary Layer ◽  
Physical Instability

AbstractIn this paper we show how the stability of Prandtl boundary layers is linked to the stability of shear flows in the incompressible Navier–Stokes equations. We then recall classical physical instability results, and give a short educational presentation of the construction of unstable modes for Orr–Sommerfeld equations. We end the paper with a conjecture concerning the validity of Prandtl boundary layer asymptotic expansions.

Prediction of Sudden Expansion Flows Using the Boundary-Layer Equations

1984 ◽  
Vol 106 (3) ◽  
pp. 285-291 ◽  
Author(s):  
O. K. Kwon ◽  
R. H. Pletcher ◽  
J. P. Lewis
Keyword(s):  
Experimental Data ◽  
Boundary Layer ◽  
Finite Difference ◽  
Stokes Equations ◽  
Computational Effort ◽  
Sudden Expansion ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Boundary Layer Equations ◽  
Difference Calculation

A finite-difference calculation method based on the boundary-layer equations is described for the prediction of laminar, developed channel flow undergoing a symmetric sudden expansion. The scheme requires only a fraction of the computational effort required for the numerical solution of the full Navier-Stokes equations that are usually employed for this flow. Predictions of the method compare very favorably with experimental data and solutions of the full Navier-Stokes equations.

Laminar boundary layers

2018 ◽  
Author(s):  
Marcel Escudier
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Boundary Layer Equations ◽  
Laminar Boundary Layers ◽  
Momentum Integral ◽  
Aerodynamic Lift

This chapter starts by introducing the concept of a boundary layer and the associated boundary-layer approximations. The laminar boundary-layer equations are then derived from the Navier-Stokes equations. The assumption of velocity-profile similarity is shown to reduce the partial differential boundary-layer equations to ordinary differential equations. The results of numerical solutions to these equations are discussed: Blasius’ equation, for zero-pressure gradient, and the Falkner-Skan equation for wedge flows. Von Kármán’s momentum-integral equation is derived and used to obtain useful results for the zero-pressure-gradient boundary layer. Pohlhausen’s quartic-profile method is then discussed, followed by the approximate method of Thwaites. The chapter concludes with a qualitative account of the way in which aerodynamic lift is generated.

On the calculation of separation bubbles

1983 ◽  
Vol 133 ◽  
pp. 287-296 ◽  
Author(s):  
Tuncer Cebeci ◽  
Keith Stewartson
Keyword(s):  
Boundary Layer ◽  
Stokes Equations ◽  
Piecewise Linear ◽  
Numerical Procedure ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Limited Region ◽  
Boundary Layer Equations ◽  
Maximum Value ◽  
Rapid Changes

The interactive boundary-layer equations for a flat plate are solved numerically when the external velocity field is piecewise linear and would provoke separation if the response of the boundary layer were neglected. A comparable problem had already been solved by Briley using the full Navier–Stokes equations. The equations are solved for various values of the Reynolds number and x0, a parameter defining the corner point of the external velocity. It is found that flows with a limited region of separation can be computed, but that, if x0 is too large, the numerical procedure breaks down. Furthermore, this maximum value is a decreasing function of R and seems to approach the value 0.12 predicted by classical theory as R → ∞. Comparison with Briley's results indicate a reasonable agreement except that different values of x0 are appropriate. It is conjectured that, once x0 increases above the acceptable maximum, rapid changes occur in the flow properties when R is large.

On laminar boundary-layer blow-off. Part 2

1971 ◽  
Vol 48 (2) ◽  
pp. 209-228 ◽  
Author(s):  
D. R. Kassoy
Keyword(s):  
Boundary Layer ◽  
Laminar Boundary Layer ◽  
Stokes Equations ◽  
Analytical Description ◽  
Navier Stokes ◽  
Large Reynolds Number ◽  
Navier Stokes Equations ◽  
Similarity Type ◽  
Boundary Layer Equations ◽  
Wall Shear

Several examples of incipient blow-off phenomena described by the compressible similar laminar boundary-layer equations are considered. An asymptotic technique based on the limit of small wall shear, and the use of a novel form of Prandtl's transposition theorem, leads to a complete analytical description of the blow-off behaviour. Of particular interest are the results for overall boundarylayer thickness, which imply that, for a given large Reynolds number, classical theory fails for a sufficiently small wall shear. A derivation of a new distinguished limit of the Navier–Stokes equations, the use of which will lead to uniformly valid solutions to blow-off type problems for Re → ∞, is included. A solution for uniform flow past a flat plate with classical similarity type injection, based on the new limit, is presented. It is shown that interaction of the injectant layers and the external flow results in a favourable pressure gradient, which precludes the classical blow-off catastrophy.

A numerical study of laminar separation bubbles using the Navier-Stokes equations

1971 ◽  
Vol 47 (4) ◽  
pp. 713-736 ◽  
Author(s):  
W. Roger Briley
Keyword(s):  
Boundary Layer ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Separation Bubble ◽  
Reynolds Numbers ◽  
Boundary Layer Theory ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Laminar Separation ◽  
Boundary Layer Equations

The flow in a two-dimensional laminar separation bubble is analyzed by means of finite-difference solutions to the Navier-Stokes equations for incompressible flow. The study was motivated by the need to analyze high-Reynolds-number flow fields having viscous regions in which the boundary-layer assumptions are questionable. The approach adopted in the present study is to analyze the flow in the immediate vicinity of the separation bubble using the Navier-Stokes equations. It is assumed that the resulting solutions can then be patched to the remainder of the flow field, which is analyzed using boundary-layer theory and inviscid-flow analysis. Some of the difficulties associated with patching the numerical solutions to the remainder of the flow field are discussed, and a suggestion for treating boundary conditions is made which would permit a separation bubble to be computed from the Navier-Stokes equations using boundary conditions from inviscid and boundary-layer solutions without accounting for interaction between individual flow regions. Numerical solutions are presented for separation bubbles having Reynolds numbers (based on momentum thickness) of the order of 50. In these numerical solutions, separation was found to occur without any evidence of the singular behaviour at separation found in solutions to the boundary-layer equations. The numerical solutions indicate that predictions of separation by boundary-layer theory are not reliable for this range of Reynolds number. The accuracy and validity of the numerical solutions are briefly examined. Included in this examination are comparisons between the Howarth solution of the boundary-layer equations for a linearly retarded freestream velocity and the corresponding numerical solutions of the Navier-Stokes equations for various Reynolds numbers.

Sign in / Sign up

Export Citation Format

Share Document