Equation stream function

Equation 9 is Laplace s equation which also occurs in several other fields of mathematical physics. Where the flow problem is two-dimensional, the velocities ate also detivable from a stream function, /. 

Superposition of Flows Potential flow solutions are also useful to illustrate the effect of cross-drafts on the efficiency of local exhaust hoods. In this way, an idealized uniform velocity field is superpositioned on the flow field of the exhaust opening. This is possible because Laplace s equation is a linear homogeneous differential equation. If a flow field is known to be the sum of two separate flow fields, one can combine the harmonic functions for each to describe the combined flow field. Therefore, if d)) and are each solutions to Laplace s equation, A2, where A and B are constants, is also a solution. For a two-dimensional or axisymmetric three-dimensional flow, the flow field can also be expressed in terms of the stream function. 

Outside the jet and away from the boundaries of the workbench the flow will behave as if it is inviscid and hence potential flow is appropriate. Further, in the central region of the workbench we expect the airflow to be approximately two-dimensional, which has been confirmed by the above experimental investigations. In practice it is expected that the worker will be releasing contaminant in this region and hence the assumption of two-dimensional flow" appears to be sound. Under these assumptions the nondimensional stream function F satisfies Laplace s equation, i.e.. 

In the case of the free jet, the solution for the Aaberg exhaust system can be found by solving the Laplace equation by the method of separation of variables and assuming that there is no fluid flow through the surface of the workbench. At the edge of the jet, which is assumed to be at 0—0, the stream function is given by Eq. (10.113). This gives rise to... 

The stream function satisfying the fourth-order differential equation, used by Haberman and Sayre (H2) is... 

If the equations of motion, the continuity relationship, and the proper stream function are properly combined, equations result which enable one to plot the flow lines for both internal and external motion. The stream function for an infinite extent of continuous phase around a single drop is (B7, H2)... 

For axisymmetric flow the species continuity equation, Eq. (1-38), written in terms of the dimensionless concentration (j) and stream function (see Chapter 1) is... 

For steady-state (no time variation) two-dimensional flows, the notion of a streamfunction has great utility. The stream function is derived so as to satisfy the continuity equation exactly. In cylindrical coordinates, there are two two-dimensional situations that are worthwhile to investigate the r-z plane, called axisymmetric coordinates, and the r-0 plane, called polar coordinates. 

Along a streamline (i.e., a line of constant ty), = 0. Equation 3.24 requires that mass flux cannot cross a streamline, since along the streamline dm = 0, and by definition, rh is the mass flow crossing the line (area). Equation 3.24 also requires that the mass flow rate between any two streamlines is related simply to the difference of the stream function on the two streamlines... 

One important use of the stream function is for the visualization of flow fields that have been determined from the solution of Navier-Stokes equations, usually by numerical methods. Plotting stream function contours (i.e., streamlines) provides an easily interpreted visual picture of the flow field. Once the velocity and density fields are known, the stream function field can be determined by solving a stream-function-vorticity equation, which is an elliptic partial differential equation. The formulation of this equation is discussed subsequently in Section 3.13.1. Solution of this equation requires boundary values for l around the entire domain. These can be evaluated by integration of the stream-function definitions, Eqs. 3.14, around the boundaries using known velocities on the boundaries. For example, for a boundary of constant z with a specified inlet velocity u(r),... 

In addition to the vorticity transport equation, a relationship between vorticity and stream function can be developed for two-dimensional steady-state problems. Continuing to use the r-6 plane as an example, the stream function is defined to satisfy the continuity equation exactly (Section 3.1.3),... 

The stream-function-vorticity equation, taken together with the vorticity transport equation, completely replaces the continuity and momentum equations. The pressure has been eliminated as a dependent variable. The continuity equation has been satisfied exactly by the stream function, and does not need to be included in the system of equations. The... 

Our intent here is not to suggest a solution method but rather to use the stream-function-vorticity formulation to comment further on the mathematical characteristics of the Navier-Stokes equations. In this form the hyperbolic behavior of the pressure has been lost from the system. For low-speed flow the pressure gradients are so small that they do not measurably affect the net pressure from a thermodynamic point of view. Therefore the pressure of the system can simply be provided as a fixed parameter that enters the equation of state. Thus pressure influences density, still accommodating variations in temperature and composition. Since the pressure or the pressure gradients simply do not appear anywhere else in the system, pressure-wave behavior has been effectively filtered out of the system. Consequently acoustic behavior or high-speed flow cannot be modeled using this approach. 

The derivation of the two-dimensional planar equations is analogous to the approach for axisymmetric coordinates. The stream function in the planar situation is... 

As with the axisymmetric stagnation-flow case, deriving the tubular stagnation-flow equations begins with the steady-state three-dimensional Navier-Stokes equations (Eqs. 3.58, 3.59, and 3.60). The approach depends on essentially the same assumptions as the axial stagnation flows described earlier, albeit with the similarity requiring no variation in the axial coordinate. The velocity field is presumed to be described in terms of a stream function that has the form... 

Notice several important aspects about the transformed equations. Of course, xjr is now an independent variable, with the relationship between r and x/r coming from the stream-function definition... 

We begin with the general boundary-layer equations (Section 7.2) where the stream function takes the form... 

Discuss the relationship between the continuity equation (Eq. 7.44) and Eq. 7.60 that represents the relationship between the physical radial coordinate and the stream function. Note that one is a partial differential equation and that the other is an ordinary differential equation. Formulate a finite-difference representation of the continuity equation in the primative form. Be sure to respect the order of the equation in the discrete representation. 

For general three-dimensional impinging streams it is difficult to define a stream function satisfying the continuity equation. However, if the velocity vector is defined by... 

S.J. Liao. Higher order stream function-vorticity formulation of 2D steady-state Navier-Stokes equation. International Journal for Numerical Methods in Fluids, 15 595-612, 1992. 

We now transform the governing equations in cylindrical coordinates into polar coordinates. Since the motion is axisymmetric, the transformation from (r, z) to (R, 6), as shown in Fig. 3.1, is analogous to the transformation from Cartesian coordinates (x, y) to cylindrical coordinates (r, 0) in a two-dimensional domain. The stream function is related to the velocity components in polar coordinates by... 

Solution of Equations in Terms of Stream Function Equation (3.9) in polar coordinates can be satisfied if 0 is in a form... 

The preceding solution is for the case involving a particle moving at a constant velocity V. For the case involving an accelerating particle, in order to obtain the stream function, we introduce the following theorem first. Let vs(r, R) be a solution of the equation... 

In the 1960s, the start of application of computers to the practice of marine research gave a pulse to the development of numerical diagnostic hydrodynamic models [33]. In them, the SLE (or the integral stream function) field is calculated from the three-dimensional density field in the equation of potential vorticity balance over the entire water column from the surface to the bottom. The iterative computational procedure is repeated until a stationary condition of the SLE (or the integral stream function) is reached at the specified fixed density field. Then, from equations of momentum balance, horizontal components of the current vector are obtained, while the continuity equation provides the calculations of the vertical component. The advantage of this approach is related to the absence of the problem of the choice of the zero surface and to the account for the coupled effect of the baroclinicity of... 

Because they do not contain the pressure as a variable. Eqs. (2.76) and (2.77) have been used quite extensively in solving problems for which the boundary layer equations (see later) cannot be used. For this purpose, instead of solving the Navier-Stokes and energy, simultaneously with the continuity equation, it is convenient to introduce the stream function, ip, which is defined such that... 

Consider two-dimensional flow over an axisymmetric body. Write the governing equations in terms of a suitably defined stream function and vorticity. 

Assuming die flow remains laminar and has a boundary layer-like characteristic, write down the governing equations together with the boundary and initial conditions. If the y coordinate is replaced by the stream function derived by ... 

The governing equations given above, i.e., Eqs. (8.122) to (8.125), are given in terms of die so-called primitive variables, i.e., u,v,p, and T. The solution procedure discussed here is based on equations involving the stream function, iff, the vorticity, o>, and the temperature, T, as variables. The stream function and vorticity are as before defined by ... 

As shown in Chapter 2, the stream function as so defined satisfies the continuity equation. 

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