Velocity interpolation

Since the computational grids are generally not coincident with the location of the particle surface, a velocity interpolation procedure needs to be carried out in order to calculate the boundary force and apply this force to the control volumes close to the immersed particle surface (Fadlun et al., 2000). 

In the IBM, the presence of the solid boundary (fixed or moving) in the fluid can be represented by a virtual body force field -rp( ) applied on the computational grid at the vicinity of solid-flow interface. Considering the stability and efficiency in a 3-D simulation, the direct forcing scheme is adopted in this model. Details of this scheme are introduced in Section II.B. In this study, a new velocity interpolation method is developed based on the particle level-set function (p), which is shown in Fig. 20. At each time step of the simulation, the fluid-particle boundary condition (no-slip or free-slip) is imposed on the computational cells located in a small band across the particle surface. The thickness of this band can be chosen to be equal to 3A, where A is the mesh size (assuming a uniform mesh is used). If a grid point (like p and q in Fig. 20), where the velocity components of the control volume are defined, falls into this band, that is... 

For this velocity, interpolation of the given data gives a value of h = 64 W/m2 K. 

In order to apply the calculation scheme outlined in the introduction the available data on p., a, and C. must be tabulated. The values selected after a critical survey of the available data are listed in Table I together with the sound velocities interpolated from our measurements. Substitution of these data into equations (1) to (7) provides the results listed in Table II. 

Using different types of time-stepping techniques Zienkiewicz and Wu (1991) showed that equation set (3.5) generates naturally stable schemes for incompressible flows. This resolves the problem of mixed interpolation in the U-V-P formulations and schemes that utilise equal order shape functions for pressure and velocity components can be developed. Steady-state solutions are also obtainable from this scheme using iteration cycles. This may, however, increase computational cost of the solutions in comparison to direct simulation of steady-state problems. 

The use of selectively reduced integration to obtain accurate non-trivial solutions for incompressible flow problems by the continuous penalty method is not robust and failure may occur. An alternative method called the discrete penalty technique was therefore developed. In this technique separate discretizations for the equation of motion and the penalty relationship (3.6) are first obtained and then the pressure in the equation of motion is substituted using these discretized forms. Finite elements used in conjunction with the discrete penalty scheme must provide appropriate interpolation orders for velocity and pressure to satisfy the BB condition. This is in contrast to the continuous penalty method in which the satisfaction of the stability condition is achieved indirectly through... 

Depending on the type of elements used appropriate interpolation functions are used to obtain the elemental discretizations of the unknown variables. In the present derivation a mixed formulation consisting of nine-node bi-quadratic shape functions for velocity and the corresponding bi-linear interpolation for the pressure is adopted. To approximate stres.ses a 3 x 3 subdivision of the velocity-pressure element is considered and within these sub-elements the stresses are interpolated using bi-linear shape functions. This arrangement is shown in Edgure 3.1. 

The momentum and continuity equations give rise to a 22 x 22 elemental stiffness matrix as is shown by Equation (3.31). In Equation (3.31) the subscripts I and / represent the nodes in the bi-quadratic element for velocity and K and L the four corner nodes of the corresponding bi-linear interpolation for the pressure. The weight functions. Nr and Mf, are bi-qiiadratic and bi-linear, respectively. The y th component of velocity at node J is shown as iPj. Summation convention on repeated indices is assumed. The discretization of the continuity and momentum equations is hence based on the U--V- P scheme in conjunction with a Taylor-Hood element to satisfy the BB condition. 

As explained in Chapter 3, it is possible to use equal order interpolation models for the spatial discretization of velocity and pressure in a U-V-P scheme based on Equations (4.127) and (4.128) without violating the BB stability condition. 

Starting point for evaluating the settling characteristics of suspended solids for dilute systems. Note that from the definition of the Reynolds number, we can readily determine the settling velocity of the particles from the application of the above expressions (u, = /xRe/dpp). The following is an interpolation formula that can be applied over all three settling regimes ... 

A priori, neither the value of O nor the values of density and velocity are known at the faces of the control volume. They have to be determined via interpolation from their values at neighboring nodes. A simple approximation would be... 

The standard deviation maps have been linearly interpolated to the same in-plane spatial resolution as the high resolution data. Images are shown for a constant gas velocity of 112 mm s and the data were recorded as a function of decreasing liquid velocity. The liquid velocities are (a) 2.8, (b) 3.7, (c) 6.1 and (d) 7.6 mm s-1. 

In the Verlet method, this equation is written by using central finite differences (see Interpolation and Finite Differences ). Note that the accelerations do not depend upon the velocities. 

The relationship between flow rate, pressure drop, and pipe diameter for water flowing at 60°F in Schedule 40 horizontal pipe is tabulated in Appendix G over a range of pipe velocities that cover the most likely conditions. For this special case, no iteration or other calculation procedures are required for any of the unknown driving force, unknown flow rate, or unknown diameter problems (although interpolation in the table is usually necessary). Note that the friction loss is tabulated in this table as pressure drop (in psi) per 100 ft of pipe, which is equivalent to 100pef/144L in Bernoulli s equation, where p is in lbm/ft3, ef is in ft lbf/lbm, and L is in ft. 

Example 11-1 Unknown Velocity and Unknown Diameter of a Sphere Settling in a Power Law Fluid. Table 11-1 summarizes the procedure, and Table 11-2 shows the results of a spreadsheet calculation for an application of this method to the three examples given by Chhabra (1995). Examples 1 and 2 are unknown velocity problems, and Example 3 is an unknown diameter problem. The line labeled Equation refers to Eq. (11-32) for the unknown velocity cases, and Eq. (11-35) for the unknown diameter case. The Stokes value is from Eq. (11-9), which only applies for - Re,pi < 1 (e.g., Example 1 only). It is seen that the solutions for Examples 1 and 2 are virtually identical to Chhabra s values and the one for Example 3 is within 5% of Chhabra s. The values labeled Data were obtained by iteration using the data from Fig. 4 of Tripathi et al. (1994). These values are only approximate, because they were obtained by interpolating from the (very compressed) log scale of the plot. 

The velocity at grid point q (in Fig. 20), which is located inside the small band and on the particle side, is determined by interpolating the fluid velocity at the neighboring grid points outside the particle surface ... 

Strictly speaking, this can only be true if the interpolated mean velocity field satisfies continuity. 

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