The Flow of a Laminar, Incompressible Jet Along a Parabola

1972 ◽  
Vol 39 (1) ◽  
pp. 13-17 ◽  
Author(s):  
A. Plotkin
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
External Flow ◽  
Boundary Layer Theory ◽  
Second Order ◽  
Wall Jet ◽  
Governing Equations ◽  
Displacement Effect ◽  
First Order ◽  
External Stream

The flow of a laminar, incompressible jet along a parabola in the absence of an external stream is analyzed using the techniques of second-order boundary-layer theory. The first-order solution is the Glauert wall-jet solution. Second-order corrections in the jet due to the effects of curvature and displacement are obtained numerically after the external flow is corrected to account for the displacement effect. The shear stress at the wall is calculated and it appears that for values of the Reynolds number at which the governing equations are valid the jet does not separate from the parabola.

An approximate theory for incompressible viscous flow past two-dimensional bluff bodies in the intermediate Reynolds number regime O(1) < Re < O(102)

1976 ◽  
Vol 77 (1) ◽  
pp. 129-152 ◽  
Author(s):  
Sheldon Weinbaum ◽  
Michael S. Kolansky ◽  
Michael J. Gluckman ◽  
Robert Pfeffer
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Pressure Gradient ◽  
Boundary Layer Theory ◽  
Navier Stokes ◽  
Bluff Bodies ◽  
Approximate Theory ◽  
Two Dimensional ◽  
First Order ◽  
Flow Past

A new approximate theory is proposed for treating the flow past smoothly contoured two-dimensional bluff bodies in the intermediate Reynolds number rangeO(1) <Re< 0(102), where the displacement effect of the thick viscous layer near the surface of the body is large and a steady laminar wake is present. The theory is based on a new pressure hypothesis which enables one to take account of the displacement interaction and centrifugal effects in thick viscous layers using conventional first-order boundary-layer equations. The basic question asked is how the wall pressure gradient in ordinary boundary -layer theory must be modified if the pressure gradient along the displacement surface using the Prandtl pressure hypothesis is to be equal to the pressure gradient along this surface using a higher-order approximation to the Navier-Stokes equation in which centrifugal forces are considered. The inclusion of the normal pressure field with displacement interaction is shown to be equivalent to stretching the streamwise body co-ordinate in first-order boundary-layer theory such that the streamwise pressure gradient as a function of distance along the original and displacement body surfaces are equal.While the new theory is of a non-rigorous nature, it yields results for the location of separation and detailed surface pressure and vorticity distribution which are in remarkably good agreement with the large body of available numerical Navier-Stokes solutions. A novel feature of the new boundary-value problem is the development of a simple but accurate approximate method for determining the inviscid flow past an arbitrary two-dimensional displacement body with its wake.

Reacting flow as an example of a boundary layer under singular external conditions

1969 ◽  
Vol 38 (3) ◽  
pp. 513-535 ◽  
Author(s):  
Raul Conti ◽  
Milton Van Dyke
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Inviscid Flow ◽  
Second Order ◽  
Order Effects ◽  
Reacting Flow ◽  
First Order ◽  
Rate Of Reaction ◽  
External Conditions ◽  
Quantitative Results

An inviscid flow with infinite gradients at the wall poses a problem for the accompanying boundary layer that is fundamentally different from the conventional one of bounded gradients. It turns out that in both cases the external gradients do not affect the first-order boundary layer, but the unbounded gradients generate second-order corrections that are of a lower order in Reynolds number than the conventional ones. The present singularity in stagnation-point reacting flow is of an algebraic type, and the boundary-layer corrections that it generates are proportional to non-integral powers of the Reynolds number, with exponents that vanish with the rate of reaction. The present example clarifies such matters as the matching of boundary layer and singular inviscid flow, the structure and decay of the new corrections, and their ranking in comparison with the conventional second-order effects. Numerical computations illustrate the problem and give quantitative results in a few selected cases.

scholarly journals Generalized second-order slip for unsteady convective flow of a nanofluid: a utilization of Buongiorno’s two-component nonhomogeneous equilibrium model

2020 ◽  
Vol 9 (1) ◽  
pp. 156-168
Author(s):  
Seyed Mahdi Mousavi ◽  
Saeed Dinarvand ◽  
Mohammad Eftekhari Yazdi
Keyword(s):  
Boundary Layer ◽  
Equilibrium Model ◽  
Second Order ◽  
Model Parameters ◽  
Slip Parameter ◽  
Two Phase ◽  
Convective Boundary ◽  
First Order ◽  
Special Cases ◽  
Two Component

AbstractThe unsteady convective boundary layer flow of a nanofluid along a permeable shrinking/stretching plate under suction and second-order slip effects has been developed. Buongiorno’s two-component nonhomogeneous equilibrium model is implemented to take the effects of Brownian motion and thermophoresis into consideration. It can be emphasized that, our two-phase nanofluid model along with slip concentration at the wall shows better physical aspects relative to taking the constant volume concentration at the wall. The similarity transformation method (STM), allows us to reducing nonlinear governing PDEs to nonlinear dimensionless ODEs, before being solved numerically by employing the Keller-box method (KBM). The graphical results portray the effects of model parameters on boundary layer behavior. Moreover, results validation has been demonstrated as the skin friction and the reduced Nusselt number. We understand shrinking plate case is a key factor affecting non-uniqueness of the solutions and the range of the shrinking parameter for which the solution exists, increases with the first order slip parameter, the absolute value of the second order slip parameter as well as the transpiration rate parameter. Besides, the second-order slip at the interface decreases the rate of heat transfer in a nanofluid. Finally, the analysis for no-slip and first-order slip boundary conditions can also be retrieved as special cases of the present model.

A theory for flow separation

1983 ◽  
Vol 132 ◽  
pp. 163-183 ◽  
Author(s):  
William S. Vorus
Keyword(s):  
Boundary Layer ◽  
Integral Equation ◽  
Reynolds Number ◽  
Circular Cylinder ◽  
Flat Plate ◽  
Linear Theory ◽  
Boundary Integral ◽  
First Order ◽  
Ideal Flow ◽  
The Ideal

This paper proposes a high-Reynolds-number theory for the approximate analysis of timewise steady viscous flows. Its distinguishing feature is linearity. But it differs fundamentally from Oseen's (1910) well-known linear theory. Oseen flow is a variation on Stokes flow at the low-Reynolds-number limit.The theory is developed for a %dimensional body moving through an infinite incompressible fluid. The velocity-vorticity formulation is employed. A boundary integral expressing the body contour velocity is written in terms of Green functions of the approximate governing differential equations. The boundary integral contains three unknown boundary distributions. These are a velocity source density, the boundary vorticity, and the normal gradient of the boundary vorticity. The unknown distributions are determined as the solutions to a boundary-integral equation formed from the velocity integral by the prescription of zero relative fluid velocity on the body boundary.The linear integral-equation formulation is applied specifically to the case of thin bodies, such that the boundary condition is satisfied approximately on the streamwise coordinate axis. The integral equation is then reduced to its leading-order contribution in the limit of infinite Reynolds number. The unknown distributions uncouple in the first-order formulation, and analytic solutions are obtained. A most interesting result appears at this point: the theory recovers linearized airfoil theory in the first-order infinite-Reynolds-number limit; the airfoil integral equation determines one of the three contour distributions sought.The first-order theory is then demonstrated by application to two classical cases: the zero-thickness flat plate at zero incidence, and the circular cylinder.For the flat plate, the streamwise velocity near the plate predicted by the proposed linear theory is compared with that of Blasius's solution to the laminar boundary-layer equations (Schlichting 1968). The linear theory predicts a fuller profile, tending more toward the character expected of the timewise steady turbulent profile. This character is also exhibited in the predicted velocity distribution across the plate wake, which is compared with Goldstein's asymptotic boundary-layer solution (Schlichting 1968). The wake defect is more severe according to the proposed theory.For the case of the circular cylinder, application of the formulation is not truly valid, since the circular cylinder is not a thin body. The theory does, however, predict that the flow separates. The separation points are predicted to lie at position angles of approximately ± 135°, with angle measured from the forward stagnation point. This compares with the prediction of 109O from the Blasius series solution to the laminar boundary-layer equations (Schlichting 1968).The theory is next applied to the case of a non-zero-thickness flat plate with incidence. From the fully attached flow at zero incidence, the theory predicts that both Ieading-edge separation and reattachment and trailing-edge separation appear on the suction side at small angle. On increasing incidence, the forward reattachment point moves aft, and the aft separation point moves forward. Coalescence occurs near midchord, and the foil is thereafter fully separated.Finally, the first-order contribution to the far-field velocity at high Reynolds number is shown to be identically that corresponding to the ideal flow. This result, coupled with the recovery of linearized thin-foil theory in the near-field limit, is argued to support strongly the physical idea that the ideal flow is, in fact, the limiting state of the complete field flow at infinite Reynolds number. Flow separation can be viewed as present in the ideal flow limit; i t is simply embedded in the infinitesimally thin body-surface vortex sheets into which the entire viscous field collapses as vorticity convection overwhelms vorticity diffusion at the infinite-Reynolds-number limit.

Experiments on Laminar Flow about a Rotating Sphere in an Air Stream

1987 ◽  
Vol 201 (6) ◽  
pp. 427-438 ◽  
Author(s):  
M A I El-Shaarawi ◽  
M M Kemry ◽  
S A El-Bedeawi
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Laminar Flow ◽  
Numerical Solutions ◽  
Separation Angle ◽  
Governing Equations ◽  
Rotating Sphere ◽  
Axis Of Rotation ◽  
Air Stream ◽  
Uniform Stream

Laminar flow about a rotating sphere which is subjected to a uniform stream of air in the direction of the axis of rotation is investigated experimentally. Measurements of the velocity components within the boundary layer and the separation angle were performed at a Reynolds number, Re, of 10 000 and Ta/Re 2 of 0, 1 and 5. These measurements are compared with the numerical solutions of the same problem where either theoretical potential or actual experimental boundary conditions are imposed on the governing equations.

scholarly journals Euler-Bernoulli Beam Theory: First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Finite Difference ◽  
Beam Theory ◽  
Second Order ◽  
Governing Equations ◽  
Deflection Curve ◽  
Order Analysis ◽  
First Order ◽  
Finite Difference Approximations ◽  
Order Polynomial ◽  
Euler Bernoulli Beam

This paper presents an approach to the Euler-Bernoulli beam theory (EBBT) using the finite difference method (FDM). The EBBT covers the case of small deflections, and shear deformations are not considered. The FDM is an approximate method for solving problems described with differential equations. The FDM does not involve solving differential equations; equations are formulated with values at selected points of the structure. Generally, the finite difference approximations are derived based on fourth-order polynomial hypothesis (FOPH) and second-order polynomial hypothesis (SOPH) for the deflection curve; the FOPH is made for the fourth and third derivative of the deflection curve while the SOPH is made for its second and first derivative. In addition, the boundary conditions and not the governing equations are applied at the beam&rsquo;s ends. In this paper, the FOPH was made for all of the derivatives of the deflection curve, and additional points were introduced at the beam&rsquo;s ends and positions of discontinuity (concentrated loads or moments, supports, hinges, springs, etc.). The introduction of additional points allowed us to apply the governing equations at the beam&rsquo;s ends and to satisfy the boundary and continuity conditions. Moreover, grid points with variable spacing were also considered, the grid being uniform within beam segments. First-order analysis, second-order analysis, and vibration analysis of structures were conducted with this model. Furthermore, tapered beams were analyzed (element stiffness matrix, second-order analysis). Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, with damping taken into account. The results obtained in this paper showed good agreement with those of other studies, and the accuracy was increased through a grid refinement. Especially in the first-order analysis of uniform beams, the results were exact for uniformly distributed and concentrated loads regardless of the grid. Further research will be needed to investigate polynomial refinements (higher-order polynomials such as fifth-order, sixth-order&hellip;) of the deflection curve; the polynomial refinements aimed to increase the accuracy, whereby non-centered finite difference approximations at beam&rsquo;s ends and positions of discontinuity would be used.

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