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.
Fluent uses Segregated Solver method to solve the governing integral equations for the conservation of mass and momentum, and (when appropriate)
33
for energy and other scalars such as turbulence and chemical species. In case a control-volume-based technique is used that consists of:
1. Division of domain into discrete control volumes using a computational grid.
2. 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.
3. Linearization of the discretized equations and solutions of the resultant linear equation system 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 time of iteration consists of the steps illustrated in Fig. 3.6 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 pressure correction equation is then solved to obtain the necessary
34
corrections to the pressure and velocity fields and the face mass fluxes 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 inter-phase 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.
In the segregated solution method each discrete governing equation is
35
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 multi-grid (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 𝜙𝜙. This is demonstrated by the following equation written in integral form for an arbitrary control volume V as follows:
∮ 𝜌𝜌𝜙𝜙𝐴𝐴⃗ ∙ 𝑑𝑑𝐴𝐴⃗ = ∮ 𝛤𝛤𝜙𝜙𝛻𝛻𝜙𝜙 ∙ 𝑑𝑑𝐴𝐴⃗ + ∮ 𝑆𝑆𝑉𝑉 𝜙𝜙𝑑𝑑𝑑𝑑 (3-18)
36
where
Eq. (3-18) is applied to each control volume, or cell, in the computational domain. The two-dimension, triangular cell shown in Fig. 3.7 is an example of such a control volume. Discretization of Eq. (3-18) on a given cell yields
∑𝑁𝑁𝑒𝑒𝑒𝑒𝑓𝑓𝑓𝑓𝑒𝑒𝑓𝑓𝜌𝜌𝑒𝑒𝐴𝐴����⃗𝜙𝜙𝑒𝑒 𝑒𝑒 ∙ 𝐴𝐴����⃗ =𝑒𝑒 ∑𝑁𝑁𝑒𝑒𝑒𝑒𝑓𝑓𝑓𝑓𝑒𝑒𝑓𝑓𝛤𝛤𝜙𝜙(𝛻𝛻𝜙𝜙)𝑖𝑖 ∙ 𝐴𝐴����⃗ + 𝑆𝑆𝑒𝑒 𝜙𝜙𝑑𝑑 (3-19)
The equations solved by Fluent take the same general form as the one given above and apply readily to multi-dimension, unstructured meshes composed of arbitrary polyhedral.
By default, Fluent stores discrete values of the scalar 𝜙𝜙 at the cell center (c0 and c1 in Fig. 3.7). However, face values 𝜙𝜙𝑒𝑒 are required for the convection
37
terms in Eq. (3-19) 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 𝜙𝜙𝑒𝑒 is set equal to the cell-center value of 𝜙𝜙 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 𝐽𝐽𝑒𝑒∗, computed from 𝐽𝐽𝑒𝑒 = 𝐽𝐽̂𝑒𝑒 + 𝑑𝑑𝑒𝑒(𝑝𝑝𝑓𝑓0 − 𝑝𝑝𝑓𝑓1) (where pc0 and pc1 are the pressures within the two cells on either side of the face, and 𝐽𝐽̂𝑒𝑒 contains the influence of velocities in these cell. The term df is a function of 𝑎𝑎𝑝𝑝, the average of the momentum equation 𝑎𝑎𝑝𝑝 coefficients for the cells on either side of face f.)
𝐽𝐽𝑒𝑒∗ = 𝐽𝐽̂∗𝑒𝑒 + 𝑑𝑑𝑒𝑒(𝑝𝑝𝑓𝑓0∗ − 𝑝𝑝𝑓𝑓1∗ ) (3-20)
does not satisfy the continuity equation. Consequently, a correction 𝐽𝐽𝑒𝑒′ is added to the face flux 𝐽𝐽𝑒𝑒∗ so that the corrected face flux, 𝐽𝐽𝑒𝑒
𝐽𝐽𝑒𝑒 = 𝐽𝐽𝑒𝑒∗ + 𝐽𝐽𝑒𝑒′ (3-21)
38
satisfies the continuity equation. The SIMPLE algorithm postulates that 𝐽𝐽𝑒𝑒′ be written as
𝐽𝐽𝑒𝑒′ = 𝑑𝑑𝑒𝑒(𝑝𝑝𝑓𝑓0′ + 𝑝𝑝𝑓𝑓1′ ) (3-22) where 𝑝𝑝′ is the cell pressure correction.
The SIMPLE algorithm substitutes the flux correction equations, Eq. (3-21) and (3-22), into the discrete continuity equation (∑𝑁𝑁𝑒𝑒𝑒𝑒𝑓𝑓𝑓𝑓𝑒𝑒𝑓𝑓𝐽𝐽𝑒𝑒𝐴𝐴𝑒𝑒 = 0) to obtain a discrete equation for the pressure correction 𝑝𝑝′ in the cell:
𝑎𝑎𝑝𝑝𝑝𝑝′ = ∑ 𝑎𝑎𝑖𝑖𝑏𝑏 𝑖𝑖𝑏𝑏𝑝𝑝𝑖𝑖𝑏𝑏′ + 𝑏𝑏 (3-23) where the source term b is the net flow rate into the cell:
𝑏𝑏 = ∑𝑁𝑁𝑒𝑒𝑒𝑒𝑓𝑓𝑓𝑓𝑒𝑒𝑓𝑓𝐽𝐽𝑒𝑒∗𝐴𝐴𝑒𝑒 (3-24) The pressure-correction equation, Eq. (3-23), may be solved using the algebraic multigrid (AMG) method. Once a solution is obtained, the cell pressure and the face flux are used correctly.
𝑝𝑝 = 𝑝𝑝∗ + 𝛼𝛼𝑝𝑝𝑝𝑝′ (3-25) 𝐽𝐽𝑒𝑒 = 𝐽𝐽𝑒𝑒∗ + 𝑑𝑑𝑒𝑒(𝑝𝑝𝑓𝑓0′ − 𝑝𝑝𝑓𝑓1′ ) (3-26) Here 𝛼𝛼𝑝𝑝 is the under-relaxation factor for pressure. The corrected face flux 𝐽𝐽𝑒𝑒 satisfies the discrete continuity equation identically during each time of 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 setup requires a means of computing the flux across the two non-conformal
39
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 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 completely intersect), as shown in Fig. 3.8. 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. 3.9, 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.