Boundary-layer flow flat-plate

It is remarkable that boundary layer flow along a flat plate, much studied and well understood, has not been used in electrochemical mass-transfer... 

Similarity Variables The physical meaning of the term similarity relates to internal similitude, or self-similitude. Thus, similar solutions in boundary-layer flow over a horizontal flat plate are those for which the horizontal component of velocity u has the property that two velocity profiles located at different coordinates x differ only by a scale factor. The mathematical interpretation of the term similarity is a transformation of variables carried out so that a reduction in the number of independent variables is achieved. There are essentially two methods for finding similarity variables, "separation of variables (not the classical concept) and the use of "continuous transformation groups. The basic theory is available in Ames (1965). 

Take into consideration two-dimensional, rectilinear, steady, incompressible, constant-property, laminar boundary layer flow in the x direction along a flat plate. Assume that viscous energy dissipation may be neglected. Write the continuity, momentum and energy equations. 

Laminar Boundary Layer Flow Along a Flat Plate with Radiation Boundary Condition... 

Consider a steady, laminar boundary layer flow of incompressible, transparent fluid along a flat plate, with a constant applied heat flux qw Btu/(hr ft2) at the wall surface. The properties of the fluid are assumed constant. The main considerations are conduction to the fluid, and radiation from the plate to the environment at Te. Surface of the plate is opaque and gray, and the uniform emissivity is 8. The fluid which is at a temperature of T,, flows at a uniform velocity of Uo. Flow velocities are sufficiently small so that viscous dissipation may be neglected. 

In the preceding sections, the solution for boundary layer flow over a flat plate wav obtained by reducing the governing set of partial differential equations to a pair of ordinary differential equations. This was possible because the velocity and temperature profiles were similar in the sense that at all values of x, (u u ) and (Tw - T)f(Tw - T > were functions of a single variable, 17, alone. Now, for flow over a flat plate, the freestream velocity, u, is independent of x. The present section is concerned with a discussion of whether there are any flow situations in which the freestream velocity, u 1, varies with Jr and for which similarity solutions can still be found [1],[10]. 

VISCOUS DISSIPATION EFFECTS ON LAMINAR BOUNDARY LAYER FLOW OVER A FLAT PLATE... 

Air flows at a velocity of 9 m/s over a wide flat plate that has a length of 6 cm in the flow direction. The air ahead of the plate has a temperature of 10°C while the surface of the plate is kept at 70°C. Using the similarity solution results given in this chapter, plot the variation of local heat transfer rate in W/m2 along the plate and the velocity and temperature profiles in the boundary layer on the plate at a distance of 4 cm from the leading edge of the plate. Also calculate the mean heat transfer rate from the plate. 

In order to measure the velocity of a stream of air, a flat plate of length 2 cm in the flow direction is placed in the flow. This plate is electrically heated, the heat dissipation rate being uniform over the plate surface. The plate is wide so a two-dimensional laminar boundary layer flow can be assumed to exist The velocity is to be deduced by measuring the temperature of the plate at its trailing edge. If this temperature is to be at least 40°C when the air temperature is 20 C and the air velocity is 3 m/s, find the required rate of teat dissipation in the plate per unit surface area. 

The conditions under which transition occurs depend on the geometrical situation being considered, on the Reynolds number, and on the level of unsteadiness in the flow well away from the surface over which the flow is occurring [2], [30]. For example, in the case of flow over a flat plate as shown in Figure 5.6, if the level of unsteadiness in the freestream flow ahead of the plate is very low, transition from laminar to turbulent boundary layer flow occurs approximately when ... 

Transition in boundary layer flow over a flat plate. 

Consider transition in the boundary layer flow over a flat plate. Using the expression for the thickness of a laminar boundary layer on a flat plate given in Chapter 3, find the value of the Reynolds number based on the boundary layer thickness at which transition begins. 

As discussed in the previous chapter, most early efforts at trying to theoretically predict heat transfer rates in turbulent flow concentrated on trying to relate the wall heat transfer rate to the wall shear stress [1],[2],[3],[41. The reason for this is that a considerable body of experimental and semi-theoretical knowledge concerning the shear stress in various flow situations is available and that the mechanism of heat transfer in turbulent flow is obviously similar to the mechanism of momentum transfer. In the present section an attempt will be made to outline some of the simpler such analogy solutions for boundary layer flows, attention mainly being restricted to flow over a flat plate. 

The effects of fluid property variations on heat transfer in turbulent boundary layer flow over a flat plate have also been numerically evaluated. This evaluation indicates that if the properties are as with If minar boundary layers evaluated at ... 

At an altitude of 30,000 m the atmospheric pressure is approximately 1200 Pa and the temperature is approximately -4S°C. Assuming a turbulent boundary layer flow over an adiabatic flat plate, plot the variation of the adiabatic wall temperature with Mach number for Mach numbers between 0 and 5. 

Available analyses of turbulent natural convection mostly rely in some way on the assumption that the turbulence structure is similar to that which exists in turbulent forced convection, see [96] to [105]. In fact, the buoyancy forces influence the turbulence and the direct use of empirical information obtained from studies of forced convection to the analysis of natural convection is not always appropriate. This will be discussed further in Chapter 9. Here, however, a discussion of one of the earliest analyses of turbulent natural convective boundary layer flow on a flat plate will be presented. This analysis involves assumptions that are typical of those used in the majority of available analyses of turbulent natural convection. 

A vertical flat plate is maintained at a uniform surface temperature and is exposed to air at standard ambient pressure. At a distance of 10 cm from the leading edge of the plate the boundary layer thickness is 2 cm. Estimate the thickness of the boundary layer at a distance of 25 cm from the leading edge. Assume a laminar boundary layer flow. 

Consider mixed convective laminar boundary layer flow over a horizontal flat plate that is heated to a uniform surface temperature. In such a flow there will be a pressure change across the boundary induced by the buoyancy forces, i.e. ... 

Merkin, J.H.. "The Effect of Buoyancy Forces on the Boundary-Layer Flow over a Semi-Infinite Vertical Flat Plate in a Uniform Free Stream , J. Fluid Mech., Vol. 35, pp.439-450, 1969. 

Fig. 5-1 Sketch showing different boundary-layer flow regimes on a flat plate...

The Grashof number may be interpreted physically as a dimensionless group representing the ratio of the buoyancy forces to the viscous forces in the free-convection flow system. It has a role similar to that played by the Reynolds number in forced-convection systems and is the primary variable used as a criterion for transition from laminar to turbulent boundary-layer flow. For air in free convection on a vertical flat plate, the critical Grashof number has been observed by Eckert and Soehngen [1] to be approximately 4 x 10". Values ranging between 10" and 109 may be observed for different fluids and environment turbulence levels. ... 

It is noted that the integral method gives only approximate values for the mass transfer coefficient as the model derivation is based on several simplifying assumptions regarding the concentration and velocity profiles. Nevertheless, the given relation has been confirmed by experiments for laminar boundary layer flows over a flat plate (e.g., [134], p 80 and p 201 [27], p 345). 

A well developed theory (12) is available to deal with simple situations flow along a flat plate, around a cylinder or sphere, over an airfoil, etc. Blunt objects such as buildings are generally handled empirically. Figure 2 depicts the perturbations created by boundary layer flows whose surface characteristics differ from those of the unperturbed atmosphere nearby or upstream ... 

Problem 10-4. Boundary Layer on a Flat Plate in an Accelerating Flow, Uoo = Ax. Consider flow past a flat plate in the throat of a 2D channel as depicted in the figure. If the free-stream velocity is given by = ax, where x is the distance from the leading edge and X is a constant, show that the flow in the boundary layer on the plate is governed by an ODE. Solve for the streamfunction numerically. How does the boundary-layer thickness grow with x How does the shear stress vary with x ... 

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