Stream function and vorticity

It is often convenient to work in terms of a dimensionless stream function and vorticity defined, respectively, as... 

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

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 ... 

At the outflow of the domain, the traditional extrapolation based on = 0 was applied for stream function and vorticity transport equations. Equations (2.7.15) and (2.7.16) have been solved in Sengupta et al. (2002) first for the case of c = Uoa, in a domain for which Reynolds number varies from 165 to 1900. The results are shown in Fig. 2.34 at different indicated times. 

To eliminate the unknown local pressure field, the set is rewritten according to the stream function and vorticity formulation [Bird et al., I960 Patankar, 1980]. This development is explained in detail in the original paper. 

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... 

At the inflow boundary and on top of the computational domain, analytic solution for the disturbance velocity was used in accordance with Eqn. (2.7.1) and (2.7.2). On the flat plate, the no-slip condition simultaneously provides a Dirichlet boundary condition for the stream function and the wall vorticity at every instant of time. 

Dynamical interpretation to the SCS circulation pattern By assuming the SCS to be an enclosed basin, the basin-scale circulation pattern can be obtained from the Sverdrup stream function. And from the Sverdrup stream function in the interior region, the mass transport of the western boundary current (WBC) is 5 6 Sv and 3 4 Sv in winter and summer, respectively. They suggested that the upper layer basin-scale circulation can be mainly regarded as a wind-driven circulation forced by the wind stress vorticity, which indicates that the SCS circulation has strong regional characteristics. 

The governing equations for the steady flow in the array in of the stream function and the vorticity are ... 

For CO 0, Eq. (11-7) reduces to the stream function for steady creeping flow past a rigid sphere, i.e., Eq. (3-7) with k = co. The parameter 3 may be regarded as a characteristic length scale for diffusion of vorticity generated at the particle surface into the surrounding fluid. When co is very large, 3 is small, and the flow can be considered irrotational except in the immediate vicinity of the particle. In the limit co go, Eq. (11-7) reduces to Eq. (1-29), the result for potential flow past a stationary sphere. 

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 two-dimensional, incompressible, steady flows, there is a relatively simple relationship between the vorticity and the stream function. Consider the axisymmetric flow as might occur in a channel, Fig. 3.12. Beginning with the axisymmetric stream function as discussed in Section 3.1.2, substitute the stream-function definition into the definition of the circumferential vorticity u>q ... 

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 boundary conditions for the stream-function-vorticity system requires specifying the stream function on all the boundaries. This is usually straightforward for known inflow and outflow conditions and solid walls. The vorticity boundary conditions comes from evaluating Eq. 3.281 on the boundary. Along the boundary, which usually corresponds with one of the coordinate directions, one of the terms in Eq. 3.281 (i.e., the one in which the derivatives align with the boundary) can be evaluated explicitly since the stream function is already specified. Thus the boundary conditions becomes a relationship between the boundary vorticity and a boundary-normal second derivative of stream function. For example, consider the natural convection in a long horizontal tube. Here, since there is no inflow or outflow, the stream function is simply zero all around the tube wall. Thus the vorticity boundary conditions are... 

The boundary conditions provide a tight coupling between the vorticity and stream-function fields. Also velocities still appear in the convective terms. Given the stream-function field, velocity is evaluated from the definition of stream function. That is, velocity is computed from stream-function derivatives. 

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. 

One troublesome aspect of solving low-speed flow problems numerically is dealing with the hyperbolic characteristics of pressure waves. Since the pressure waves usually have no importance in these problems, they are mainly a mathematical and computational nuisance. Therefore techniques to filter pressure waves are often desirable. The stream-function-vorticity approach accomplishes this filtering, but it is not now used very much... 

Consider the condition, which determines the velocity of the curved flame front propagation in the channel. Inside the stagnation zone filled by combustion products the pressure is constant and is equal to the value at infinity (when x = oo). Because of Bernoulli s integral along the streamline restricting the stagnation zone, the gas motion velocity remains unchanged. Since at x = oo the flow is plane-parallel (ptJO = const, v — 0), distributions of velocity u and of the stream function are associated with the vorticity distribution ... 

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... 

The two-dimensional Navier-Stokes equation is solved in stream function-vorticity formulation, as reported variously in Sengupta et al. (2001, 2003), Sengupta Dipankar (2005). Brinckman Walker (2001) also simulated the burst sequence of turbulent boundary layer excited by streamwise vortices (in X- direction) using the same formulation for which a stream function was defined in the y — z) -plane only. To resolve various small scale events inside the shear layer, the vorticity transport equation (VTE) and the stream function equation (SFE) are solved in the transformed — rj) —... 

Figure 3.8 Stream function (top three panels) and vorticity contours plotted at the indicated times. Same contour values are plotted for each quantity. Arrowheads at the top of each frame indicate the instantaneous streamwise location of the freestream...

Equations (1) and (2) are said to be in primitive Ux, Uy, u, p) variable form. While there are other forms that rely on derived variables, such as stream function vorticity, vorticity-velocity potential, and dual-potential methods, we restrict ourselves to the primitive variable form because of their popularity and ease of interpretation. For a discussion of these methods, as well as for further details of most aspects of CFD discussed here, see Refs. 73, 74, and 77. 

Potential Flow around a Gas Bubble Via the Stream Function. The same axisymmetric flow problem in spherical coordinates is solved in terms of the stream function All potential flow solutions yield an intricate network of equipotentials and streamlines that intersect at right angles. For two-dimensional ideal flow around a bubble, the velocity profile in the preceding section was calculated from the gradient of the scalar velocity potential to ensure no vorticity ... 

One solves for the stream function by invoking no vorticity at the microscopic level. Since Vr and vg are both functions of r and 9, with = 0, the r and 9 components of the vorticity vector are ffivially zero. The (/>-component of (V X v) yields an equation that must be solved for ir(r,9). Hence, one combines the nontrivial component of the vorticity vector with the relations between Vr, vg and f, given by (8-239) ... 

It is left as an exercise for the student to verify that these relations between the two nonzero velocity components and ilr conserve overall fluid mass, and that streamlines intersect equipotentials at right angles in cylindrical coordinates. The stream function is obtained by invoking no vorticity at the microscopic level. Only the z-component of the fluid vorticity vector yields nontrivial information about ir. For example. 

Answer Approach (i) requires that one must solve the ( -component of the equation of motion. Since is a function of both r and 9, there are two nonzero components of V x v (i.e., the r and 9 components are nontrivial). Hence, approach (ii) requires that one must consider the r and 9 components of the Laplacian of the vorticity vector to obtain an expression for the stream function via the low-Reynolds-number equation of change for fluid vorticity. The preferred approach is (1). 

Another technique, widely used in a different context (kinetic theory, plasma simulation) is the so-called Vortex-in-Cell Method (Hockney and Eastwood 1981, Birdsall and Langdon 1985). In the Vortex-in-Cell Method a two-dimensional computational domain is divided into cells and the vorticity is counted in each cell. The Poisson equation relating the vorticity and the Laplacian of the stream function is subsequently solved on the grid by an FTT algorithm, the velocity... 

In employing the B.E.M. to the problem in hand, a dimensionless stream function-vortic-ity formulation has been used. Results are presented for constant stream function values in the vicinity of the inlet to a thrust pad enabling a visualisation of the flow pattern. A determination of the pressure variation on the moving surface (or runner) of the bearing in the inlet region is also undertaken. The non-zero value of pressure at the nominal inlet to the pad is detailed with consideration of both Poiseuille and Couette dominated flows. 

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