Prandtls Boundary-Layer Equations

Ludwig Prandtl, the father of boundary-layer theory, after making the conceptual division of the flow discussed in Sec. 10.1, set out to calculate the flow in the boundary layer. He chose as his starting point the Navier-Stokes equations (Sec. 7.9) and simplified them by dropping the terms he considered unimportant. His simplifications are as follows  

The solid surface is taken as the x axis, the boundary layer beginning at the origin see Fig. 11.1. 

Gravity is unimportant compared with the other forces acting, so the gravity term can be dropped. 

The flow is two-dimensional in the x and y directions. This means that (the z component of the velocity) is zero, as are all derivatives with respect to 2. These simplifications make the z momentum balance all zeros, so it can be dropped from the list of equations to be solved. 

Although therel is some flow in the y direction within the boundary layer, it is slow enough compared with the flow in the x direction for us not to need to consider the y-directed momentum balance. This does not mean that Vy is zero, but it does mean that dPIdy is negligible. 

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See also -> convection, -> Grashof number, - Hagen-Poiseuille, -> hydrodynamic electrodes, -> laminar flow, - turbulent flow, -> Navier-Stokes equation, -> Nusselt number, -> Peclet number, -> Prandtl boundary layer, - Reynolds number, -> Stokes-Einstein equation, -> wall jet electrode. 

These relations are extremely valuable as they slate that for a given geometry, the friction coefficient can be expres.sed as a function of Reynolds number alone, and the Nusselt number as a function of Reynolds and Prandtl numbers alone (Fig. 6-34). Therefore, experimentalists can study a problem with a minimum number of experiments, and report their friction and heat transfer coefficient measure.ments conveniently in terms of Reynolds and Prandtl numbers. For example, a friction coefficient relation obtained with air for a given surface can also be used for water at the same Reynolds number. But it should be kepi in mind that the validity of these relations is limited by the limitations on (he boundary layer equations used in the analysis. 

For large Reynolds numbers fie — oo all terms with the factor 1 /fie and even more those with 1/fie2 will be vanishingly small. After neglecting these terms and returning to equations with dimensions, we obtain the boundary layer equations, which were first given in this form by L. Prandtl (1875-1953) [3.4] in 1904 ... 

As soon as the functional relationships between the Nusselt, Reynolds and Prandtl numbers or the Sherwood, Reynolds and Schmidt numbers have been found, be it by measurement or calculation, the heat and mass transfer laws worked out from this hold for all fluids, velocities and length scales. It is also valid for all geometrically similar bodies. This is presuming that the assumptions which lead to the boundary layer equations apply, namely negligible viscous dissipation and body forces and no chemical reactions. As the differential equations (3.123) and (3.124) basically agree with each other, the solutions must also be in agreement, presuming that the boundary conditions are of the same kind. The functions (3.126) and (3.128) as well as (3.127) and (3.129) are therefore of the same type. So, it holds that... 

As the previous illustrations showed, the heat and mass transfer coefficients for simple flows over a body, such as those over flat or slightly curved plates, can be calculated exactly using the boundary layer equations. In flows where detachment occurs, for example around cylinders, spheres or other bodies, the heat and mass transfer coefficients are very difficult if not impossible to calculate and so can only be determined by experiments. In terms of practical applications the calculated or measured results have been described by empirical correlations of the type Nu = f(Re,Pr), some of which have already been discussed. These are summarised in the following along with some of the more frequently used correlations. All the correlations are also valid for mass transfer. This merely requires the Nusselt to be replaced by the Sherwood number and the Prandtl by the Schmidt number. 

The analysis for Sc -> oo closely follows the large-Prandtl-number analysis of Section C. The solution of (11-113) is a singular perturbation with a leading-order outer solution 0 = 0 and an inner solution that satisfies a boundary-layer equation that is obtained by rescaling according to... 

Blasius steady-flow, laminar, flat-plate, boundary-layer solution is a numerical solution of his simplification of Prandtl s boundary-layer equations, which are a simplified, one-dimensional momentum balance and a mass balance. This type of solution is known in the boundary-layer literature as an exact solution. Exact solutions can be found for only a very limited number of cases. Therefore, approximate methods are available for making reasonable estimates of the behavior of laminar boundary layers (Prob. 11.8). 

Prandtl started with the Navier-Stokes equations and discarded enough terms to make his boundary-layer equations, which are the working form of the momentum and continuity equations for boundary-layer problems. 

The above equation indicates that c is a constant following a fluid particle. Suppose we have c = Cq in the free stream and the wall boundary condition for a reacting surface is = 0. The solution of equation (4.75) cannot satisfy the boundary conditions at the reaction surface. Evidently, near the surface, there must be a thin diffusion boundary layer of thickness djy within which the concentration changes rapidly (see Figure4.15). This reasoning parallels the Prandtl boundary layer argument for viscous flow past a solid boundary at high Reynolds number. If is the Prandtl viscous boundary layer thickness for steady unbounded laminar flow, we know that... 

The velocity of the fluid may be assumed to obey the Prandtl one seventh power law, given by equation 11.26. If the boundary layer thickness S is replaced by the pipe radius r, this is then given by ... 

In developing the boundary layer concept, Prandtl suggested an order-of-magnitude evaluation of the terms in the Navier-Stokes equations, which provides an expression (with an unknown proportionality constant) for the... 

There is a natural draw rate for a rotating disk that depends on the rotation rate. Both the radial velocity and the circumferential velocity vanish outside the viscous boundary layer. The only parameter in the equations is the Prandtl number in the energy equation. Clearly, there is a very large effect of Prandtl number on the temperature profile and heat transfer at the surface. For constant properties, however, the energy-equation solution does not affect the velocity distributions. For problems including chemistry and complex transport, there is still a natural draw rate for a given rotation rate. However, the actual inlet velocity depends on the particular flow circumstances—there is no universal correlation. 

A comparison of Eqs. (3.52) and (3.53) and also of their boundary conditions as given in Eqs. (3.24) and (3.54) respectively, shows that these equations are identical in all respects. Therefore, for the particular case of Pr equal to one, the distribution of 9 through the boundary layer is identical to the distribution of uJu ). In this par-ticular case, therefore, Fig. 3.4 also gives the temperature distribution and the two boundary layer thicknesses are identical in this case. Now many gases have Prandtl numbers which are not very different from 1 and this relation between the velocity and temperature fields and the results deduced from it will be approximately correct for them. 

If terms of the order (SIL) and less are again neglected and itf it is assumed that the Prandtl number, Pr, has order 1 or greater, i.e., is not small,j it follows that the x-wise diffusion term, i.e., d2T/dx2. is negligible compared to the other terms. Hence, the energy equation for free convective laminar two-dimensional boundary layer flow becomes ... 

It will be seen from the results given by the similarity solution that the velocities are very low in natural convective boundary layers in fluids with high Prandtl numbers. In such circumstances, the inertia terms (i.e., the convective terms) in the momentum equation are negligible and the boundary layer momentum equation for a vertical surface effectively is ... 

Liquid metals such as mercury have high thermal conductivities, and are commonly used in applications that require high heat transfer rates. However, they have very small Prandtl numbers, and thus the thermal boundary layer develops much faster than the velocity boundary layer. Then we can assume the velocity in the thermal boundary layer to be constant at the free stream value and solve the energy equation. It gives... 

The influence of a wall on the turbulent transport of scalar (species or enthalpy) at the wall can also be modeled using the wall function approach, similar to that described earlier for modeling momentum transport at the wall. It must be noted that the thermal or mass transfer boundary layer will, in general, be of different thickness than the momentum boundary layer and may change from fluid to fluid. For example, the thermal boundary layer of a high Prandtl number fluid (e.g. oil) is much less than its momentum boundary layer. The wall functions for the enthalpy equations in the form of temperature T can be written as ... 

In order to calculate wx/wT by solving the differential equation (3.149), the Reynolds stress w xw has to be known. The hypothesis introduced by Boussinesq (3.140) is unsuitable for this, as according to it, the Reynolds stress does not disappear at the wall. However, the condition w xw y = 0 at the wall is satisfied by Prandtl s mixing length theory, which will now be explained. In order to do this we will consider a fluid element in a turbulent boundary layer, at a distance y from the wall, Fig. 3.16. It has, at a distance y, the mean velocity wx(y) and... 

A boundary layer flow is one of the simplest two-dimensional formulations in fluid mechanics. This approach suggested by Prandtl assumes that the flow properties change very slowly in the longitudinal direction with respect to the normal direction, Jj. This allows one to drop some terms in the equations, thus having simplified them. 

The previous problem made use of the Prandtl s boundary layer approach. Although it is widely applied, it would be worth to examine this approximation on a test problem, where the complete Navier-Stokes equations should be solved. Here is such a test problem that has also its own significance. 

The existing turbulence models consist of approximate relations for the /ij-parameter in (5.246). The Prandtl mixing-length model (1.356) represents an early algebraic (zero-equation) model for the turbulent viscosity Ht in turbulent boundary layers. 

However, the Blasius function f(rj) is available only as the numerical solution of the Blasius equation, and it is thus inconvenient to evaluate this formula for H(r] ). A simpler alternative is to numerically integrate the Blasius equation and the thermal energy equation (11-19) simultaneously. The function H(r]), obtained in this manner, is plotted in Fig. 11-2 for several different values of the Prandtl number, 0.01 < Pr < 100. As suggested earlier, it can be seen that the thermal boundary-layer thickness depends strongly on Pr. For Pr 1, the thermal layer is increasingly thin relative to the Blasius layer (recall that / ->. 99 for rj 4). The opposite is true for Pr <[Pg.773]

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