Two-Dimensional Bodies

For 2D body shapes, h2 = 1. In addition either h2/hx = 1 (for example elliptical cylindrical, bipolar, or parabolic cylindrical coordinate22), or h2/hx = 1 + 0(Pe xli) (for circular cylindrical coordinates assuming that r = 1 is the surface of the cylinder). Hence (9 254) simplifies to the universal form, at least for all 2D geometries for which there is a known analytic coordinate system  

In the case of a solid sphere, considered in the preceding section, we solved the thermal boundary-layer equation analytically by using a similarity transformation. An obvious question is whether we may also solve (9-257) by means of the same approach. To see whether a similarity solution exists, we apply a similarity transformation of the form 

The constant coefficient that appears in (9-259) and (9-260) is arbitrary. The value 3 is chosen for convenience. To determine whether a similarity solution exists, we must determine whether solutions of (9-259) and (9-260) can be found that satisfy appropriate boundary conditions. Specifically, to satisfy boundary condition on 0 for Y = 0 and the matching condition as Y - oo, we require that the similarity function satisfies 

In addition, g(q2) is required to be finite except possibly at values of x corresponding to thermal wakes (for example, at the downstream stagnation point in the case of a sphere) for which the assumption of a thin thermal layer is no longer valid. 

(9-260) is identical to equation (9 234), which was found earlier for the sphere, and we have already seen that it can be solved subject to the conditions (9 261). The solution for 9 is given in (9 240). The existence of a similarity solution to (9 257) thus rests with Eq. (9 259). Specifically, for a similarity solution to exist, it must be possible to obtain a solution of (9-259) for g(q2), which remains finite for all q2 except possibly at a stagnation point where a = 0, from which a thermal wake may emanate. 

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Suppose that a body is strongly elongated in some direction and with sufficient accuracy it can be treated as the two-dimensional. In other words, an increase of the dimension of the body in this direction does not practically change the field at the observation points. We will consider a two-dimensional body with an arbitrary cross section and introduce a Cartesian system of coordinates x, y, and z, as is shown in Fig. 4.5a, so that the body is elongated along the y-axis. It is clear that if at any plane y — constant the behavior of the field is the same. To carry out calculations we will preliminarily perform two procedures, namely,... 

Fig. 4.5. (a) Field due to two-dimensional body, (b) field caused by two elementary masses, (c) the field due to thin two-dimensional layer. 

Each elementary prism is replaced by an infinitely thin line directed along the y-axis with the same mass per unit length as that of the prism. These two steps allow us to replace the two-dimensional body by a system of infinitely thin lines which are parallel to each other, and the distribution of mass on them is defined from the equality... 

Here pairs Xg, Zg and Xp, Zp are Cartesian coordinates of any point of a body and observation points, respectively. Equation (4.17) describes the vertical component of the field caused by masses within an elementary prism. Correspondingly, for the field g, due to a two-dimensional body, we have... 

To further simplify calculations consider the special case when a two-dimensional body is oriented along the y-axis, and its thickness is much smaller than the distance between the body and observation points that is, h< observation point. Now Equation (4.18) is greatly simplified and we have... 

Now we will show that, making use of this equation, it is possible to calculate the gravitational field caused by masses in a two-dimensional body with an arbitrary cross section. With this purpose in mind, let us mentally divide the body cross section into a sufficient number of relatively thin layers with the thickness hi. Then, applying the principle of superposition and Equation (4.21) for elementary layer, we have... 

The objective of any heat-transfer analysis is usually to predict heat flow or the temperature which results from a certain heat flow. The solution to Eq. (3-1) will give the temperature in a two-dimensional body as a function of the two independent space coordinates x and y. Then the heat flow in the x and y directions may be calculated from the Fourier equations... 

Consider a two-dimensional body which is to be divided into equal increments in both the x and y directions, as shown in Fig. 3-5. The nodal points are designated as shown, the m locations indicating the x increment and the n locations indicating the y increment. We wish to establish the temperatures at any of these nodal points within the body, using Eq. (3-1) as a governing condition. Finite differences are used to approximate differential-increments in the temperature and space coordinates and the smaller we choose these finite increments, the more closely the true temperature distribution will be approximated. 

Consider a two-dimensional body divided into increments as shown in Fig. 4-19. The subscript m denotes the x position, and the subscript n denotes the y position. Within the solid body the differential equation which governs the heat flow is... 

A useful two-dimensional expression for concentration of a chemical introduced as a pulse over the depth of a vertically mixed layer of water is given by Eq. [2-18], (The equation includes the effect of first-order decay.) This equation is a solution to Eq. [1-6] under conditions of an instantaneous injection of mass into an infinite two-dimensional body of water,... 

Figure 6.28. Drag coefficient vs. Reynolds number for two-dimensional bodies [49]...

Cone in Supersonic Flow. The preceding solutions for a flat plate may be applied to a cone in supersonic flow through the Mangier transformation [39], which in its most general form relates the boundary layer flow over an arbitrary axisymmetric body to an equivalent flow over a two-dimensional body. This transformation is contained in Eq. 6.89, which results in transformed axisymmetric momentum and energy equations equivalent to the two-dimensional equations (Eqs. 6.95 and 6.97). Hence, solutions of these equations are applicable to either a two-dimensional or an axisymmetric flow, the differences being contained solely in the coordinate transformations. 

The skin friction coefficient and Stanton number are defined under the conditions of similarity on axisymmetric and two-dimensional bodies as follows. The components of the shear stress in the xt direction are given by... 

The lines of flow direction marked A and B in Fig. 10.8 close on the point at which there is zero flow (that is, y = 0 and x = — CIA). If we divide the flow along eurve Us and consider only the flow to the left of this curve, this represents tlie potential flow outside some two-dimensional body shaped like AB.H the flow is from left to right, then this is quite similar to the flow over the leading edge of an airplane wing or the upstream side of a rounded bridge abutment. 

It should be noticed that in the previous considerations we write x and x for the placement of the material particle and its velocity independent of the dimension of the body, that is, whether the body is three-dimensional. This kind of notation will be used later as well except when one-dimensional or two-dimensional continuum is observed. It is the case, for instance, when the material surface is contained in three-dimensional body. In that case x and x are quantities which are related to two-dimensional body. With = (H, t) we denote the position of the material particle of three-dimensional material body. Also with... 

Having in mind that dm = gAV and dmo- = yda, where g and y are mass density of three- and two-dimensional bodies, respectively, we write... 

M. Isaacson and K. E. Cheung, Time-domain solution for wave-current interactions with a two-dimensional body, Appl. Ocean Res. 15, 39-52 (1993). 

J.F. Thompson, Z.U.A. Warsi, and C.W. Mastin (1974) Automatic numerical generation of body-fitted curvilinear coordinate system for field containing any number of arbitrary two-dimensional bodies. J. Comput. Phys. 15, 299-319. J.F. Thompson, Z.U.A. Warsi, and C.W. Mastin (1985) Numerical Grid Generation. North-Holland, New York. 

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