An analytical solution of the Navier-Stokes equations for unsteady backward stagnation-point flow with injection or suction

2006 ◽  
Vol 86 (4) ◽  
pp. 281-290 ◽  
Author(s):  
Alan Shapiro
Keyword(s):  
Analytical Solution ◽  
Stagnation Point ◽  
Stokes Equations ◽  
Stagnation Point Flow ◽  
Navier Stokes ◽  
Navier Stokes Equations

Oscillating stagnation point flow

1982 ◽  
Vol 384 (1786) ◽  
pp. 175-190 ◽  
Keyword(s):  
Boundary Layer ◽  
Stagnation Point ◽  
High Frequency ◽  
Stokes Equations ◽  
Stagnation Point Flow ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Analytic Approximations ◽  
Numerical Integrations ◽  
Hiemenz Flow

A solution of the Navier-Stokes equations is given for an incompressible stagnation point flow whose magnitude oscillates in time about a constant, non-zero, value (an unsteady Hiemenz flow). Analytic approximations to the solution in the low and high frequency limits are given and compared with the results of numerical integrations. The application of these results to one aspect of the boundary layer receptivity problem is also discussed.

scholarly journals Unsteady Reversed Stagnation-Point Flow over a Flat Plate

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Vai Kuong Sin ◽  
Chon Kit Chio
Keyword(s):  
Laminar Flow ◽  
Flow Field ◽  
Asymptotic Solution ◽  
Stagnation Point ◽  
Stokes Equations ◽  
Stagnation Point Flow ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Total Flow ◽  
Navier Stokes Equations

This paper investigates the nature of the development of two-dimensional laminar flow of an incompressible fluid at the reversed stagnation-point. Proudman and Johnson (1962) first studied the flow and obtained an asymptotic solution by neglecting the viscous terms. Robins and Howarth (1972) stated that this is not true in neglecting the viscous terms within the total flow field. Viscous terms in this analysis are now included, and a similarity solution of two-dimensional reversed stagnation-point flow is investigated by solving the full Navier-Stokes equations.

On the Immersed Boundary Method: Finite Element Versus Finite Volume Approach

2012 ◽  
Author(s):  
Angelo Frisani ◽  
Yassin A. Hassan
Keyword(s):  
Finite Element ◽  
Analytical Solution ◽  
Finite Volume ◽  
Stokes Equations ◽  
Numerical Results ◽  
Navier Stokes ◽  
Immersed Boundary ◽  
Order Accuracy ◽  
Navier Stokes Equations ◽  
Two Dimensional Flow

A projection approach is presented for the coupled system of time-dependent incompressible Navier-Stokes equations in conjunction with the Immersed Boundary Method (IBM) for solving fluid flow problems in the presence of rigid objects not represented by the underlying mesh. The IBM allows solving the flow for geometries with complex objects without the need of generating a body fitted mesh. The no-slip boundary constraint is satisfied applying a boundary force at the immersed body surface. Using projection and interpolation operators from the fluid volume mesh to the solid surface mesh (i.e., the “immersed” boundary) and vice versa, it is possible to impose the extra constraint to the incompressible Navier-Stokes equations as a Lagrange multiplier in a fashion very similar to the effect pressure has on the momentum equations to satisfy the divergence-free constraint. The projection operation removes the immersed boundary surface slip and non-divergence-free components of the velocity field. The boundary force is determined implicitly at the inner iterations of the fractional step method implemented. No constitutive relations for the immersed boundary objects fluid interaction are required, allowing the formulation introduced to use larger CFL numbers compared to previous methodologies. An overview of the immersed boundary approach is presented showing third order accuracy in space and second order accuracy in time when the simulation results for the Taylor-Green decaying vortex are compared to the analytical solution using the Immersed Finite Element Method (IFEM). For the Immersed Finite Volume Method (IFVM) a ghost-cell approach is used. Second order accuracy in space and first order accuracy in time are obtained when the Taylor-Green decaying vortex test case is compared to the analytical solution. The numerical results are compared with the analytical solution also for adaptive mesh refinement (for the IFEM) showing an excellent error reduction. Computations were performed using IFEM and IFVM approaches for the time-dependent incompressible Navier-Stokes equations in a two-dimensional flow past a stationary circular cylinder at Re = 20, and 40, where shedding effects are not present. The drag coefficient and the recirculation length error compared to the experimental data is less than 3–4%. Simulations for the two-dimensional flow past a stationary circular cylinder at Re = 100 were also performed. For Re numbers above 46, unsteadiness generates vortex shedding, and an unsteady flow regime is present. The results shown are in excellent quantitative and qualitative agreement with the flow pattern expected. The numerical results obtained with the discussed IFEM and IFVM were also compared against other immersed boundary methodologies available in literature and simulation performed with the commercial computational fluid dynamics code STAR-CCM+/V5.02.009 for which a body fitted finite volume numerical discretization was used. The benchmark showed that the numerical results obtained with the implemented immersed boundary methods are very close to those obtained from STAR-CCM+ with a very fine mesh and in a good agreement with the other IBM techniques. The IBM based of finite element approach is numerically more accurate than the IBM based on finite volume discretization. In contrast, the latter is computationally more efficient than the former.

scholarly journals A semi-analytical solution for the mean wind profile in the Atmospheric Boundary Layer: the convective case

2009 ◽  
Vol 9 (5) ◽  
pp. 19817-19844
Author(s):  
L. Buligon ◽  
G. A. Degrazia ◽  
O. C. Acevedo ◽  
C. R. P. Szinvelski ◽  
A. G. O. Goulart
Keyword(s):  
Boundary Layer ◽  
Analytical Solution ◽  
Wind Profile ◽  
Stokes Equations ◽  
Integral Transform ◽  
Integral Transforms ◽  
Navier Stokes ◽  
Convective Boundary ◽  
Navier Stokes Equations ◽  
Average Wind

Abstract. A novel methodology to derive the average wind profile from the Navier-Stokes equations is presented. The development employs the Generalized Integral Transform Technique (GITT), which joints series expansions with Integral Transforms. The new approach provides a solution described in terms of the quantities that control the wind vector with height. Parameters, such as divergence and vorticity, whose magnitudes represent sinoptic patterns are contained in the semi-analytical solution. The results of this new method applied to the convective boundary layer are shown to agree with wind data measured in Wangara experiment.

MHD FLOWS DUE TO NON-COAXIAL ROTATIONS OF POROUS DISK AND A VISCOUS FLUID AT INFINITY: GRAPHICAL SOLUTIONS USING MATLAB

2020 ◽  
Vol 9 (11) ◽  
pp. 9287-9301
Author(s):  
R. Lakshmi ◽  
Santhakumari
Keyword(s):  
Velocity Field ◽  
Analytical Solution ◽  
Viscous Fluid ◽  
Stokes Equations ◽  
Daily Life ◽  
Vital Role ◽  
Navier Stokes ◽  
Porous Disk ◽  
Navier Stokes Equations ◽  
Complex Phenomena

Fluids play a vital role in many aspects of our daily life. We drink water, breath air, fluids run through our bodies and it controls the weather. The study of motion of fluids is a complex phenomena. The equations which govern the flows of Newtonian fluids are Navier-Stokes equations. In this paper, the flows which are due to non – coaxial rotations of porous disk and a fluid at infinity are considered. Analytical solution for the velocity field using Laplace transform is derived. MATLAB coding is written to get the graphical solutions. The results are compared with the existing results. MATLAB software provides accurate results depending on the solution we obtained.

Laminar film flow on a cylindrical surface

1976 ◽  
Vol 74 (2) ◽  
pp. 297-315 ◽  
Author(s):  
Ernst Becker
Keyword(s):  
Stagnation Point ◽  
Cylindrical Surface ◽  
Film Flow ◽  
Stokes Equations ◽  
Inviscid Flow ◽  
Navier Stokes ◽  
Point Problem ◽  
Navier Stokes Equations ◽  
Laminar Film ◽  
Particular Solution

The paper deals with steady laminar film flow which is set up at the cylindrical surface of an idealized horizontal ‘road’ when homogeneous ‘rain’ is falling onto the road in a vertical downward direction. It is shown that a particular solution of the Navier-Stokes equations is possible for which the depth of the liquid film is constant. In that case the Navier-Stokes equations reduce to the equations governing plane stagnation-point flow. However, the boundary conditions differ from those for the classical stagnation-point problem. Solutions for nearly inviscid flow and predominantly viscous flow are derived analytically. In particular, simple formulae for the depth of the film are found in both cases. Finally, the importance of the particular solution as a member of a whole class of solutions is discussed on the basis of a momentum integral approximation.

Nonaxisymmetric Three-Dimensional Stagnation-Point Flow and Heat Transfer on a Flat Plate

2009 ◽  
Vol 131 (7) ◽  
Author(s):  
Ali Shokrgozar Abbassi ◽  
Asghar Baradaran Rahimi
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Flat Plate ◽  
Stagnation Point ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Stagnation Point Flow ◽  
Navier Stokes ◽  
Dimensional Flow ◽  
Flow And Heat Transfer

The existing solutions of Navier–Stokes and energy equations in the literature regarding the three-dimensional problem of stagnation-point flow either on a flat plate or on a cylinder are only for the case of axisymmetric formulation. The only exception is the study of three-dimensional stagnation-point flow on a flat plate by Howarth (1951, “The Boundary Layer in Three-Dimensional Flow—Part II: The Flow Near Stagnation Point,” Philos. Mag., 42, pp. 1433–1440), which is based on boundary layer theory approximation and zero pressure assumption in direction of normal to the surface. In our study the nonaxisymmetric three-dimensional steady viscous stagnation-point flow and heat transfer in the vicinity of a flat plate are investigated based on potential flow theory, which is the most general solution. An external fluid, along z-direction, with strain rate a impinges on this flat plate and produces a two-dimensional flow with different components of velocity on the plate. This situation may happen if the flow pattern on the plate is bounded from both sides in one of the directions, for example x-axis, because of any physical limitation. A similarity solution of the Navier–Stokes equations and energy equation is presented in this problem. A reduction in these equations is obtained by the use of appropriate similarity transformations. Velocity profiles and surface stress-tensors and temperature profiles along with pressure profile are presented for different values of velocity ratios, and Prandtl number.

Approximate Analytical Solution for Laminar Flow Over a Backward Step

2015 ◽  
Author(s):  
P. Venkataraman
Keyword(s):  
Analytical Solution ◽  
Stokes Equations ◽  
Approximate Analytical Solution ◽  
Continuous Functions ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Mesh Free ◽  
Numeric Calculation ◽  
Domain Discretization

Analytical solution of Navier-Stokes equations are extremely difficult and rare. It is one of the unsolved Clay Millennium problems in mathematics. Many solutions that exist are examples of degenerate cases where the nonlinearity is controlled. In this paper we explore the application of Bézier functions to solve the two-dimensional laminar fluid flow over a backward step. The Bézier functions provide a mesh free alternative to domain discretization methods that are currently used to solve such problems. The Navier-Stokes equation are handled directly without transformation and the setup is direct, simple, and involves minimizing the error in the residuals of the differential equations along with the error on the boundary conditions over the domain. The solutions for the velocity and pressure are available in polynomial form. They are single continuous functions over the entire domain. The procedure employs a combination of symbolic and numeric calculation in MATLAB. Two problems are explored. The first is the flow in a 2D channel to illustrate the technique. The second is the flow over the backward step. The solutions are compared to the corresponding finite element solutions from COMSOL Multiphysics software.

Mixed Convection Stagnation-Point Flow Past a Vertical Flat Plate With a Second Order Slip

2013 ◽  
Vol 136 (1) ◽  
Author(s):  
Alin V. Roşca ◽  
Ioan Pop
Keyword(s):  
Mixed Convection ◽  
Stagnation Point ◽  
Stokes Equations ◽  
Reverse Flow ◽  
Flow Region ◽  
Second Order ◽  
Navier Stokes ◽  
Convection Flow ◽  
Lower Branch ◽  
Navier Stokes Equations

An exact similarity solution of the steady mixed convection flow of a viscous and incompressible fluid in the vicinity of two-dimensional stagnation-point with a second-order slip condition has been investigated. Using appropriate similarity variable, the Navier–Stokes equations coupled with the energy equation governing the flow and heat transfer are reduced to a system of nonlinear ordinary (similarity) equations, which are well-posed. These equations are solved numerically in the buoyancy assisting and opposing flow regions. It is found that a reverse flow region develops in the buoyancy opposing flow case, and dual (upper and lower branch) solutions are found to exist in the case of opposing flow region for a certain range of the negative values of the mixed convection parameter. A stability analysis has been performed, which shows that the upper branch solutions are stable and physically realizable in practice, while the lower branch solutions are not stable and, therefore, not physically realizable in practice. The numerical results have been compared with those reported in the literature, the agreement being excellent.

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