This study employs the computational fluid dynamics software Fluent to analyze the flow fields around rotating Savonius wind rotors. The finite volume iteration and SIMPLE algorithm are put in use to solve the governing equations of a transient flow field. And the corresponding grid movement is also solved by using sliding mesh method.
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FLUENT uses Segregated Solver method to solve the governing integral equations for the conservation of mass and momentum, and (when appropriate) for energy and other scalars such as turbulence and chemical species. In case a control-volume-based technique is used that consists of:
Division of the domain into discrete control volumes using a computational grid.
Integration of the governing equations on the individual control volumes to construct algebraic equations for the discrete dependent variables such as velocities, pressure, temperature, and conserved scalars.
Linearization of the discretized equations and solution of the resultant linear equation system to yield updated values of the dependent variables.
3.5.1 Segregated Solution Method
Using this approach, the governing equations are solved sequentially (i.e., segregated from one another). Because the governing equations are non-linear (and coupled), several iterations of the solution loop must be performed before a converged solution is obtained. Each iteration consists of the steps illustrated in Fig. 3.5 and outlined below:
1. Fluid properties are updated, based on the current solution. (If the calculation has just begun, the fluid properties will be updated based on the initialized solution.)
2. The u, v, and w momentum equations are each solved in turn using current values for pressure and face mass fluxes, in order to update the velocity field.
3. Since the velocities obtained in Step 2 may not satisfy the continuity equation locally, a Poisson-type equation for the pressure correction is derived from the continuity equation and the linearized momentum equations. This
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pressure correction equation is then solved to obtain the necessary corrections to the pressure and velocity fields and the face mass fluxes such that continuity is satisfied.
4. Where appropriate equations for scalars such as turbulence, energy, species, and radiation are solved using the previously updated values of the other variables.
5. When interphase coupling is to be included, the source terms in the appropriate continuous phase equations may be updated with a discrete phase trajectory calculation.
6. A check for convergence of the equation set is made.
These steps are continued until the convergence criteria are met.
3.5.2 Linearization: Implicit
In the segregated solution method the discrete, non-linear governing equations are linearized to produce a system of equations for the dependent variables in every computational cell. The resultant linear system is then solved to yield an updated flow-field solution.
The manner in which the governing equations are linearized may take an implicit form with respect to the dependent variable (or set of variables) of interest.
The implicit form is described in the following:
Implicit: For a given variable, the unknown value in each cell is computed using a relation that includes both existing and unknown values from neighboring cells. Therefore each unknown will appear in more than one equation in the system, and these equations must be solved simultaneously to give the unknown quantities.
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In the segregated solution method each discrete governing equation is linearized implicitly with respect to that equation's dependent variable. This will result in a system of linear equations with one equation for each cell in the domain. Because there is only one equation per cell, this is sometimes called a scalar system of equations. A point implicit (Gauss-Seidel) linear equation solver is used in conjunction with an algebraic multigrid (AMG) method to solve the resultant scalar system of equations for the dependent variable in each cell. For example, the x-momentum equation is linearized to produce a system of equations in which u velocity is the unknown. Simultaneous solution of this equation system (using the scalar AMG solver) yields an updated u-velocity field.
In summary, the segregated approach solves for a single variable field (e.g., p) by considering all cells at the same time. It then solves for the next variable field by again considering all cells at the same time, and so on. There is no explicit option for the segregated solver.
3.5.3 Discretization
FLUENT uses a control-volume-based technique to convert the governing equations to algebraic equations that can be solved numerically. This control volume technique consists of integrating the governing equations about each control volume, yielding discrete equations that conserve each quantity on a control-volume basis.
Discretization of the governing equations can be illustrated most easily by considering the steady-state conservation equation for transport of a scalar quantity ϕ. This is demonstrated by the following equation written in integral form for an arbitrary control volume V as follows:
34 domain. The two-dimension, triangular cell shown in Fig. 3.6 is an example of such a control volume. Discretization of Eq. (3-24) on a given cell yields
Nf faces ρfv ϕf f ∙ A =f Nf faces Γϕ(∇ϕ)n ∙ A + Sf ϕV (3-25) given above and apply readily to multi-dimension, unstructured meshes composed of arbitrary polyhedral.
By default, FLUENT stores discrete values of the scalar ϕ at the cell center (c0 and c1 in Fig. 3.6). However, face values ϕf are required for the
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convection terms in Eq. (3-25) and must be interpolated from the cell center values. This is accomplished using an upwind scheme.
First-Order Upwind Scheme
When first-order accuracy is desired, quantities at cell faces are determined by assuming that the cell-center values of any field variable represent a cell-average value and hold throughout the entire cell; the face quantities are identical to the cell quantities. Thus when first-order upwind is selected, the face value ϕf is set equal to the cell-center value of ϕ in the upstream cell.
3.5.4 SIMPLE Algorithm
The SIMPLE algorithm uses a relationship between velocity and pressure corrections to enforce mass conservation and to obtain the pressure field.
If the momentum equation is solved with a guessed pressure field p*, the resulting face flux Jf∗, computed from Jf = J f + df(pc0 − pc1) (where pc0 and pc1
are the pressures within the two cells on either side of the face, and J f contains the influence of velocities in these cell. The term df is a function of ap, the average of the momentum equation ap coefficients for the cells on either side of face f.)
Jf∗ = J ∗f + df(pc0∗ − pc1∗ ) (3-26) does not satisfy the continuity equation. Consequently, a correction Jf′ is added to the face flux Jf∗ so that the corrected face flux, Jf
Jf = Jf∗ + Jf′ (3-27) satisfies the continuity equation. The SIMPLE algorithm postulates that Jf′
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be written as
Jf′ = df(pc0′ + pc1′ ) (3-28) where p′ is the cell pressure correction.
The SIMPLE algorithm substitutes the flux correction equations, Eq. (3-27) and (3-28), into the discrete continuity equation ( Nf faces JfAf = 0) to obtain a discrete equation for the pressure correction p′ in the cell:
app′ = anb nbpnb′ + b (3-29) where the source term b is the net flow rate into the cell:
b = Nf faces Jf∗Af (3-30) The pressure-correction equation, Eq. (3-29), may be solved using the algebraic multigrid (AMG) method. Once a solution is obtained, the cell pressure and the face flux are used correctly.
p = p∗ + αpp′ (3-31) Jf = Jf∗+ df(pc0′ − pc1′ ) (3-32) Here αp is the under-relaxation factor for pressure. The corrected face flux Jf satisfies the discrete continuity equation identically during each iteration.
3.5.5 Sliding Mesh
The sliding mesh model allows adjacent grids to slide relative to one another. In doing so, the grid faces do not need to be aligned on the grid interface. This situation requires a means of computing the flux across the two non-conformal interface zones of each grid interface.
To compute the interface flux, the intersection between the interface zones is determined at each new time step. The resulting intersection produces one
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interior zone (a zone with fluid cells on both sides) and one or more periodic zones. If the problem is not periodic, the intersection produces one interior zone and a pair of wall zones (which will be empty if the two interface zones intersect entirely), as shown in Fig. 3.7. The resultant interior zone corresponds to where the two interface zones overlap; the resultant periodic zone corresponds to where they do not. The number of faces in these intersection zones will vary as the interface zones move relative to one another. Principally, fluxes across the grid interface are computed using the faces resulting from the intersection of the two interface zones, rather than from the interface zone faces themselves.
In the example shown in Fig. 3.8, the interface zones are composed of faces A-B and B-C, and faces D-E and E-F. The intersection of these zones produces the faces a-d, d-b, b-e, etc. Faces produced in the region where the two cell zones overlap (d-b, b-e, and e-c) are grouped to form an interior zone, while the remaining faces (a-d and c-f) are paired up to form a periodic zone. To compute the flux across the interface into cell IV, for example, face D-E is ignored and faces d-b and b-e are used instead, bringing information into cell IV from cells I and III, respectively.