# General CFD FAQ

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## Contents |

## Physics

### What is the Reynolds decomposition?

The Reynolds decomposition is used to separate the scales in a turbulent flow and resolve the velocity field (or any other scalar field) as the sum of an average component and a fluctuating component. The time average of the fluctuating field is identically zero.

### What is the difference between the Reynolds decomposition and LES filtering?

The Reynolds decomposition separates the velocity field into an average component and a fluctuating component. **The time average of the fluctuating component is identically zero**. In LES, a filtering operation is performed to decompose the velocity field into a *filtered* velocity field and a *filtered residual* field. The crucial difference between both methods is

- The filtered velocity field is a
**random variable**while the average field (Reynolds) is not - The time average of the residual field is generally
**nonzero**while it is identically zero for fluctuating component in the Reynolds decomposition.

If the DNS is modelled, can we get the steady state solution?

## Numerics

### What is the difference between FEM, FVM and FDM?

A finite difference method (FDM) discretization is based upon the differential form of the PDE to be solved. Each derivative is replaced with an approximate difference formula (that can generally be derived from a Taylor series expansion). The computational domain is usually divided into hexahedral cells (the grid), and the solution will be obtained at each nodal point. The FDM is easiest to understand when the physical grid is Cartesian, but through the use of curvilinear transforms the method can be extended to domains that are not easily represented by brick-shaped elements. The discretization results in a system of equation of the variable at nodal points, and once a solution is found, then we have a discrete representation of the solution.

A finite volume method (FVM) discretization is based upon an integral form of the PDE to be solved (e.g. conservation of mass, momentum, or energy). The PDE is written in a form which can be solved for a given finite volume (or cell). The computational domain is discretized into finite volumes and then for every volume the governing equations are solved. The resulting system of equations usually involves fluxes of the conserved variable, and thus the calculation of fluxes is very important in FVM. The basic advantage of this method over FDM is it does not require the use of structured grids, and the effort to convert the given mesh in to structured numerical grid internally is completely avoided. As with FDM, the resulting approximate solution is a discrete, but the variables are typically placed at cell centers rather than at nodal points. This is not always true, as there are also face-centered finite volume methods. In any case, the values of field variables at non-storage locations (e.g. vertices) are obtained using interpolation.

A finite element method (FEM) discretization is based upon a piecewise representation of the solution in terms of specified basis functions. The computational domain is divided up into smaller domains (finite elements) and the solution in each element is constructed from the basis functions. The actual equations that are solved are typically obtained by restating the conservation equation in weak form: the field variables are written in terms of the basis functions, the equation is multiplied by appropriate test functions, and then integrated over an element. Since the FEM solution is in terms of specific basis functions, a great deal more is known about the solution than for either FDM or FVM. This can be a double-edged sword, as the choice of basis functions is very important and boundary conditions may be more difficult to formulate. Again, a system of equations is obtained (usually for nodal values) that must be solved to obtain a solution.

Comparison of the three methods is difficult, primarily due to the many variations of all three methods. FVM and FDM provide discrete solutions, while FEM provides a continuous (up to a point) solution. FVM and FDM are generally considered easier to program than FEM, but opinions vary on this point. FVM are generally expected to provide better conservation properties, but opinions vary on this point also. If you are trying to decide which method to use, then the best path is probably found by consulting the literature in the specific problem area.

### How do I count the number of floating point operations in an algorithm?

First, write down your algorithm as pseudocode. Then simply count the number of arithmetic operations that the algorithm is performing (per iteration if the algorithm loops). All operations (addition, subtraction, multiplication, and division) are usually counted as the same, i.e. **1 flop**. This is not exactly true, since multiplication includes several additions and division includes several multiplications when actually executed by a computer. However, we are looking for an estimate here, so it is reasonable to assume that on average, all operations count in the same manner.

Here is an example (just for illustration):

- for i = 0 to P

- for n = 1 to N (number of elements in array)

- B(n) = a(n)*a(n-1) - 2*c(n) + 3

- B(n) = a(n)*a(n-1) - 2*c(n) + 3
- next n

- for n = 1 to N (number of elements in array)
- next i

For each **n** there are 2 multiplications, 1 subtraction, and 1 addition resulting in **4** operations. This loop is executed **N** times, so there are **4N** operations. This is the the order of the algorithm. In this example, its is O(4N) or simply O(N) (constants do not count).

For all iterations, there are 4N(P+1) operations. (remember, when starting the loop from 0 to P, there are P+1 steps).