Modified Theory of Laminar Boundary Layer Flow Over a Flat Plate (original) (raw)
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A Realistic Theoretical Model for Laminar Flow over a Flat Plate
The problem of theoretically describing forced laminar flow over a flat plate is revisited. For the last hundred years it has been assumed that the Blasius solution model applies to this case. However, upon close review it is found that the Blasius model has a serious problem in that the Blasius model assumes that the pressure gradient on the plate in the flow direction is zero. In fact what one expects is that a pressure gradient develops as fluid is displaced from the plate due to the developing boundary layer. Therefore, the Blasius model is not a valid physical model of the flow over a flat plate as depicted in most textbooks. In this report , we develop a more realistic Falkner-Skan type similar solution that closely matches the flow one would expect for flow over a flat plate. We replace the usual zero pressure gradient assumption with a nonzero pressure gradient assumption. The resulting solution for the velocity profile parallel to the plate results in a velocity profile that is very similar to the Blasius solution velocity profile. The big difference in the two models is in the velocity perpendicular to the plate. For the Blasius solution , this velocity results in a net outflow whereas the new model's velocity results in a net inflow. This net inflow is critical in that it allows one to use the flat plate as a model for a wing with aerodynamic lift.
2008
The solution of the modified form of the equation of motion is compared with the classical Blasius solution, exact numerical solution of the Navier-Stokes equation, and the experimental observations of Burgers and Zijnen (1924), Hansen (1930), Nikuradse (1942), Dhawan (1953), and Büttner and Czarske (2005) for laminar boundary layer flow over a flat plate. Nearly all of the experimental data, except some of the data of Nikuradse, obtained by three different velocity measurement techniques of hot wire anemometry, Pitot tubes, and laser Doppler interferometry are in close agreement with the modified theory and show small but systematic deviation from the classical theory. Also, the predicted friction coefficient of the modified theory is found to be in closer agreement with the experimental observations of Janour (1951) as compared to that of the Navier-Stokes equation of motion.
Laminar Boundary Layer Flow of a Non-Newtonian Power Law Fluid Past a Moving Flat Plate
2016
The steady, two dimensional laminar boundary layer flow of nonNewtonian power-law fluid passing through a continuously moving flat plate under the influence of transverse magnetic field is analyzed. The non-linear partial differential equations governing the flow are transformed into a nonlinear ordinary differential equation using appropriate similarity transformations. This nonlinear ordinary differential equation is linearized by using Quasilinearization technique and then solved numerically by using implicit finite difference scheme. The system of algebraic equations is solved by using Thomson algorithm. The solution is found to be dependent on various governing parameters including magnetic field parameter M, power-law index n and velocity ratio parameter ε. A systematical study is carried out to illustrate the effects of these major parameters on the velocity profiles. It is found that dual solutions exists when the plate and the fluid move in opposite directions, near the reg...
Journal of Mathematics Research, 2012
We apply second, third, fourth, fifth and sixth order velocity profiles to discuss laminar boundary layer flow over a flat plate. Inclusion of these velocity profiles in Von Karman-Pohlhausen (1921) momentum integral equation enables us to determine the approximate values of the parameters namely, (i) boundary layer thickness, (ii) displacement thickness, (iii) momentum thickness, (iv) thickness ratio, (v) skin friction coefficient, (vi) drag coefficient and (vii) the shear rate relation on the plate. Comparison of the approximate values with the exact Blasius (1908) values leads to the determination of the percentage error for each of the above parameters for the different velocity profiles. From the sixth order velocity profile we can predict that higher order velocity profiles will yield greater percentage errors and hence worse and worse results for these parameters except displacement thickness.
Laminar Flow Analysis over a Flat Plate by Computational Fluid Dynamics 756-764
In this paper theoretical estimations of boundary layer thickness and heat transfer coefficient is examined using Computational Fluid Dynamics (CFD) for laminar air flow. The feasibility and accuracy of using CFD to calculate convective heat transfer coefficients is examined. A grid sensitivity analysis is performed for the CFD solutions, and it is used to determine the convective heat transfer coefficients. The coefficients are validated using analytical solution. In addition the local Nusselt number are obtained, which can be used in estimation of flow and heat transfer performance over a flat plate. The results tell us that for the laminar forced convection simulations the convective heat transfer coefficients differed from analytical values by 5%. The result also tells us that the boundary-layer thickness for laminar flow decreases with distance from the leading edge of the flat plate and increases with Reynolds number. The effect of Reynolds number, Prandtl number on flow is also investigated. These estimations can quickly give us the conclusion of dependences between the variables of interest.
A Simple Model for Turbulent Boundary Layer Momentum Transfer on a Flat Plate
Chemical Engineering & Technology, 2000
A simple model is presented for turbulent momentum transfer on a flat plate. The proposed model is based on some polynomial velocity profiles in a laminar sublayer as well as in a fully developed boundary layer and two integral boundary layer equations. The model could be used for the calculation of boundary layer thickness, velocity profile and skin friction factor on the flat plate. The calculated results are in very good agreement with other proposed empirical correlations.
MHD boundary layer flow of a non-newtonian power-law fluid on a moving flat plate
The problem of steady, two-dimensional laminar flow of a power-law fluid passing through a moving flat plate under the influence of transverse magnetic field is studied. The resulting governing partial differential equation is transformed into a non linear ordinary differential equation using appropriate transformation. This non linear ordinary differential equation is linearized by using Quasi-linearization technique and then solved numerically by using implicit finite difference scheme. The system of algebraic equations is solved by using Gauss-Seidal iterative method. The solution is found to be dependent on various governing parameters including magnetic field parameter M, power-law index n and velocity ratio parameter ε. A systematical study is carried out to illustrate the effects of these major parameters on the velocity profiles. It is found that dual solutions exits when the plate and the fluid move in opposite directions, near the region of separation.
Ethiopian Journal of Education and Sciences, 2018
Analysis of steady laminar boundary layer flow past a moving plate in a viscous incompressible fluid with the presence of thermal radiation has been presented in this paper. The cases when the plate and the fluids moves in the same direction and reverse to each other with species concentration are considered for the present study. The governing partial differential equations are transformed into ordinary differential equations using similarity transformations, which are more convenient for numerical computations. The transformed ordinary differential equations are then solved numerically by the Keller box method. Some numerical results obtained are compared with previously reported cases available from the literature and they are found to be in a good agreement. In addition to this, a parametric study is performed in the investigation to illustrate the influence of various parameters on the velocity, temperature and concentration profiles. Keywords: Boundary layer, moving plate, mass dif...
Radiation and Pressure Gradient Effects on the Incompressible Laminar Boundary Layer Flow
In fluid mechanics and aerospace engineering radiation and pressure gradient effects have a significant role in flow separation and heat transfer. In this study, we utilize classical perturbation theory and semi-analytical techniques for ordinary differential equations (ODEs) to obtain semi-analytical solutions of the problem under consideration. The partial differential equations (PDEs) of the problem are the continuity, the Navier-Stokes and the energy equations. We assume incompressible and laminar flow past a flat plate with pressure gradient and radiation parameter. Using the dimensionless Falkner-Skan transformation we obtain a non-linear and coupled system of two PDEs, that has a parabolic nature. By using a perturbation method, keeping terms of order up to ε 2 , the system of the two PDEs is transformed into a system of six ODEs. In this study, we use two semi-analytical techniques, the Homotopy Analysis Method (HAM) and the Differential Transformation Method (DTM) to solve the ODE system. The results are compared with the numerical solution of the ODE system. Adverse pressure gradient decreases the dimensionless velocity of the boundary layer. Favorable pressure gradient have the opposite effect for the dimensionless velocity profile. The combination of adverse pressure gradient and radiation increase the temperature of the boundary layer in the case of a cooling plate and decrease the temperature in the case of a heating plate.
On laminar boundary-layer blow-off. Part 2
Journal of Fluid Mechanics, 1971
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 catast...