4.2 Concepts of the Boundary Layer Theory

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1 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell 4.2 Concepts of the Boundary Layer Theory It is difficult to solve the complete viscous flow fluid around a body unless the geometry is very simple. Full domain numerical solution is also time consuming and not practical, because one needs to solve the full Navier-Stokes equations in full domain, which are also nonlinear, elliptic, and complex. 1

influence of viscosity is observed in a very thin domain, close to the object as shown in

2 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Prandtl Theory In 1904 Prandtl discovered that for most practical applications, the influence of viscosity is observed in a very thin domain, close to the object as shown in Figure 4.2. Therefore, outside this region one can assume the flow is inviscid (µ=0) (Prandtl, 1904; Schlichting and Gersteu, 2000; White, 2005). 2

3 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell The thin region where the effect of viscosity is dominant is called the momentum or viscous boundary layer. The solution of boundary layer analysis can be simplified due to the fact that its thickness is much smaller than the characteristic dimension of the object. The fluid adjacent to the surface of the body has zero relative velocity, u fluid u body = 0, at the surface. This is also called the no slip boundary condition. 3

4 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell One of the assumptions of boundary layer approximation is that the fluid velocity is at rest relative to the surface. This is true, except when the fluid pressure is very low and therefore Kn / L of fluid molecules is much larger than 1. In general, the flow next to an object can be divided in two parts. The larger part is related to a free stream of fluid, in which the effect of viscosity is not important (potential flow theory). The smaller region is a thin layer next to the surface of the body, in which the effects of molecular l transport t (such as viscosity, it thermal conductivity and mass diffusivity) are very important. 4

5 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Potential flow theory neglects the effect of viscosity and therefore significantly simplifies the Navier-Stokes equation, which provides the solution of the velocity distribution. A disadvantage of potential theory is that since second order terms are neglected, the effect of viscosity, no slip and impermeability boundary condition at the surface can not be accounted for. In general, potential flow theory predicts the free stream field accurately, despite its simplicity. Boundary layer thickness is defined as the distance that most of the velocity change is occurring. This thickness is usually defined as the thickness in which the velocity reaches 99% of the free stream velocity u = 0.99U. 5

6 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell Figure 4.3 illustrates how the momentum boundary layer thickness changes along the plate. Flow is laminar at relatively l small values of x where it is shown as the laminar boundary layer region. At layer value of x, the fluid motion begins to fluctuate. This is called the transition region. The boundary layer may be either laminar or turbulent in this region. Finally at a given value of x, the flow will be always turbulent. There is a very thin region next to the wall in the turbulent region that this flow is still laminar which is call laminar sublayer. 6

7 Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell For flow over a flat plate, experimental data indicates Re x the flow is laminar where Re x <Re <3 6 x 10 the flow is in transition U x Re x the flow is turbulent Laminar region Transition region Turbulent region U Laminar sublayer x δ Figure 4.3 Laminar and turbulent flow over a flat plate 7

CONVECTIVE HEAT TRANSFER

CONVECTIVE HEAT TRANSFER Mohammad Goharkhah Department of Mechanical Engineering, Sahand Unversity of Technology, Tabriz, Iran CHAPTER 3 LAMINAR BOUNDARY LAYER FLOW LAMINAR BOUNDARY LAYER FLOW Boundary