Laminar Boundary Layers and Their Separation From Curved Surfaces

1965 ◽  
Vol 87 (2) ◽  
pp. 483-493 ◽  
Author(s):  
B. S. Massey ◽  
B. R. Clayton
Keyword(s):  
Stokes Equations ◽  
Equation Of Motion ◽  
Navier Stokes ◽  
Curved Surfaces ◽  
Integration Technique ◽  
Navier Stokes Equations ◽  
Laminar Boundary Layers ◽  
Displacement Thickness ◽  
Order Of Magnitude ◽  
Similar Solutions

The equation of motion in a laminar boundary layer on curved surfaces has been developed from the Navier-Stokes equations. After an order-of-magnitude analysis terms of highest and next order have been retained and a single ordinary differential equation of motion derived by the method of “similar solutions.” An iteration procedure has been used to determine effects of displacement thickness and surface curvature up to and including separation. The equation of motion has been solved by a digital computer using the Runge-Kutta step-by-step integration technique.

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.

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 FIRST PROJECTION OF THE EQUATION OF MOTION IN THE CYLINDRICAL COORDINATE SYSTEM

2016 ◽  
pp. 90-92
Author(s):  
A. G. Obukhov ◽  
R. E. Volkov
Keyword(s):  
Coordinate System ◽  
Stokes Equations ◽  
Equation Of Motion ◽  
Cylindrical Coordinate System ◽  
Navier Stokes ◽  
Gas Flows ◽  
Heat Conducting ◽  
Complex Flows ◽  
Navier Stokes Equations ◽  
Cylindrical Coordinate

It is proved that complex flows of the viscous compressible heat-conducting gas, arising during heating the vertical field, have a pronounced axial symmetry. Therefore, for the numerical solution of the full Navier-Stokes equations for description of such gas flows it are advisable to use a cylindrical coordinate system. This paper describes the transformation of the first projection of the equation of motion of the full Navier-Stokes equations system. The result of the transformation is a record of the first projection of the equation of a continuous medium motion in the cylindrical coordinate system.

Flow induced by jets and plumes

1981 ◽  
Vol 108 ◽  
pp. 55-65 ◽  
Author(s):  
W. Schneider
Keyword(s):  
Stokes Equations ◽  
Outer Region ◽  
Reynolds Numbers ◽  
Slip Condition ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Laminar Jet ◽  
Order Of Magnitude ◽  
Valid Solution ◽  
Solid Walls

The order of magnitude of the flow velocity due to the entrainment into an axisymmetric, laminar or turbulent jet and an axisymmetric laminar plume, respectively, indicates that viscosity and non-slip of the fluid at solid walls are essential effects even for large Reynolds numbers of the jet or plume. An exact similarity solution of the Navier-Stokes equations is determined such that both the non-slip condition at circular-conical walls (including a plane wall) and the entrainment condition at the jet (or plume) axis are satisfied. A uniformly valid solution for large Reynolds numbers, describing the flow in the laminar jet region as well as in the outer region, is also given. Comparisons show that neither potential flow theory (Taylor 1958) nor viscous flow theories that disregard the non-slip condition (Squire 1952; Morgan 1956) provide correct results if the flow is bounded by solid walls.

scholarly journals Numerical modelling of bank failures during reservoir draw-down

2018 ◽  
Vol 40 ◽  
pp. 03001 ◽  
Author(s):  
Nils Reidar B. Olsen ◽  
Stefan Haun
Keyword(s):  
Stokes Equations ◽  
Limit Equilibrium ◽  
Numerical Algorithms ◽  
High Viscosity ◽  
Navier Stokes ◽  
Bank Failures ◽  
Navier Stokes Equations ◽  
Order Of Magnitude ◽  
Sediment Layers ◽  
Hydropower Reservoir

Numerical algorithms are presented for modeling bank failures during reservoir flushing. The algorithms are based on geotechnical theory and the limit equilibrium approach to find the location and the depth of the slides. The actual movements of the slides are based on the solution of the Navier-Stokes equations for laminar flow with high viscosity. The models are implemented in the SSIIM computer program, which also can be used for modelling erosion of sediments from reservoirs. The bank failure algorithms are tested on the Bodendorf hydropower reservoir in Austria. Comparisons with measurements show that the resulting slides were in the same order of magnitude as the observed ones. However, some scatter on the locations were observed. The algorithms were stable for thick sediment layers, but instabilities were observed for thin sediment layers.

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

Layered flow in slider bearings

1998 ◽  
Vol 212 (1) ◽  
pp. 47-54 ◽  
Author(s):  
M. B. W. Nabhan ◽  
G. A. Ibrahim ◽  
T. L. Whomes ◽  
M. Z. Anabtawi
Keyword(s):  
Stokes Equations ◽  
Interface Layer ◽  
Navier Stokes ◽  
Integration Technique ◽  
Slider Bearing ◽  
Total Flow ◽  
Load Bearing Capacity ◽  
Navier Stokes Equations ◽  
One Dimensional ◽  
Layered Flow

A procedure for solving the Navier-Stokes equations for the steady, one-dimensional flow of a binary water-based lubricant within an infinite slider bearing is described. The method uses an iterative process (Newton-Raphson procedure) to obtain the interface layer elevation between the liquid layers. The implementation of the Simpson rule integration technique into the equation set allows pressures along the bearing, drag on the moving bearing face, total flow into the bearing and the load-bearing capacity for a binary lubricant to be traced. Results are obtained for a range of non-linearity factors and lead to the conclusion that all the important indicators of bearing performance can be determined using the technique described.

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