^{1}

^{*}

^{2}

^{*}

Most numerical transient flow models that consider dynamic friction employ a finite differences approach or the method of characteristics. These models assume a single fluid (water only) with constant density and pressure wave velocity. But when transient flow modeling attempts to integrate the presence of air, which produces a variable density and pressure-wave velocity, the resolution scheme becomes increasingly complex. Techniques such as finite volumes are often used to improve the quality of results because of their conservative form. This paper focuses on a resolution technique for unsteady friction using the Godunov approach in a finite volume method employing single-equivalent twophase flow equations. The unsteady friction component is determined by taking into account local and convective instantaneous accelerations and the sign of both convective acceleration and velocity values. The approach is used to reproduce a set of transient flow experiments reported in the literature, and good agreement between simulated and experimental results is found.

Friction in pressurized flows can be decomposed into two components: static friction in steady flows and dynamic friction in transient flows. Transient flows present fairly important dynamic frictions that can significantly modify system behavior. Several types of model are used to calculate the dynamic friction component. One of the first types is the convolution-based model developed by Zielke [

with:

where is the Brunone coefficient, is the pipe diameter, is the water velocity, is the pressure wave celerity, is the time and is the abscissa.

Bergant et al. [

For solving transient flows with the dynamic friction, several numerical methods are proposed in the literature and two approaches are generally used. The first approach involves calculating the local and convective acceleration in the source term (see Equation (4)) as treated by Bergant et al. [

Finite volume techniques, particularly those using the Godunov method (as presented by Toro [

The first section of the paper proposes the governing equations and the second a numerical method adapted from Guinot’s [

The pressurized flow Equations (3) and (4) are those presented by Guinot [

Vectors, and correspond, respectively, to the variables, flux and source term defined by Equation (4).

where: and represent the time and the abscissa, the flow cross-section, the flow discharge, the fluid density, the mass of fluid per unit length of the pipe, the mass discharge, the pressure, the pipe slope, the friction slope calculated by Dary-Weisbach formula as following, acceleration of gravity, and the water velocity.

The friction coefficient is determined by considering local and convective instantaneous accelerations as in the model by Brunone et al. [8,9] that was modified by Bergant et al. [

where the Brunone coefficient depends on expressed in Equation (6).

Friction is therefore first decomposed into static and dynamic components as shown in Equation (5). The dynamic friction component is transformed according to the variables flow and in Equation (7):

In which:

The Brunone coefficient is determined as in Bergant et al. [

where (Equation ) depends more on the flow regime (Reynolds number).

The contributions of the dynamic friction acceleration in the vector variable and in the flux vector (Equation (4)) allows to obtain Equation (11), in which is the friction slope due to the dynamic friction component.

In vectorial form Equation (11) become Equation (12), where the sought variables are and. The other variables will be determined by considering the Equations (24), (25) and.

with:

and

Equations (12) and (13) are equivalent to Equations (3) and (4) if the dynamic friction component is not considered (i.e.,).

The next proposes a numerical method for solving these equations for each internal cell. The exterior cells will be determined by the boundary conditions formalized in each case. Since a variable is known for each pipe end, one of the Riemann invariants will be used to calculate the second variable. For example, for a known discharge or pressure to the left boundary, the second Riemann invariant is used to calculate the second variable (discharge or pressure). When it is the right boundary, the first Riemann invariant is used. Details of the calculation will be discussed in the next section.

Equations (12) and (13) resemble the Guinot’s [

The first step is to make the solution of the Riemann problem of Equation (15) to express the conservation component of Equation (12):

The second is to analyze the solution of the Riemann problem of the second hyperbolic part (Equation (16)):

The third is to obtain the solution of the Riemann problem of the source term corresponding to Equation (17):

The fourth step is to obtain all parameters (as velocity and pressure in each cell) needed to determine the flow. The next time step is calculated by taking into account the Courant condition:, corresponding to the spatial step.

The conservative part is close to the two-phase flow of the equation presented by Guinot [

After processing and arrangement, the eigenvalues of the matrix are given by Equation (19).

With:

The eigenvalues, corresponding to the eigenvectors for the matrix, are and as seen in Equation (21).

The Riemann invariants along each characteristic are given by Equation (22).

Equation (22) can be integrated respectively between and following approximation according to the trapezoidal rule, as with static friction. represents the variables to the left of the region of the constant state, the region of the constant state and those to the right of the constant state region. The resulting equation (Equation (23)) is solved by the Newton Raphson method for determining i.e. and.

Note that the celerity, which depends on, is related to other parameters through Equation (24), as presented by Guinot [

With fluid density, the volume fraction of air at the reference pressure, the polytropic coefficient of air (for isothermal conditions and for adiabatic conditions). The pressure waves celerity in water is calculated as suggested by Wylie and Streeter [

The solution of Equation (23) yields: and i.e. and used for calculating the mass discharge and the vector variable. The pressure force is obtained by solving Equation (25) [

The approach assumes half-virtual cells at and as suggested by Guinot [

If the pressure is prescribed, i.e. known at the left boundary, then Equation (25) is used to determine. Therefore Equation (24) can be used to determine from. The Riemann invariant along (Equation (23)) yields the velocity(Equation (26)) with corresponds to flow parameter at cell 1 at time.

If the discharge is known, then the velocityRiemann invariant (Equation (27) alongis combined with Equations (24) and (25) yields and.

If the pressure is prescribed at the right boundary, Equations (25) and (24), and Riemann invariant along (Equation (28) or (29)) provide and with corresponding to flow parameter at cell at time.

When discharge is prescribed, Riemann invariant (Equation (28)) along, combined with Equations (24) to (25), yields, and as shown in Equation (30):

The conservative part yields the homogeneous solution (Equation (31)) at each cell without the source term as presented by Guinot [

The flux will be calculated by Equation (32):

The provisional values will be corrected in two successive stages, integrating the second hyperbolic part and the source term.

Equation (16) is solved by seeking the eigenvalues and vectors of (Equations (33)).

Given that eigenvalues and vectors of differ depending on the sign of, each case of Equations (33) is treated separately in sections 4.2.1 for and in section 4.2.2 for.

Eigenvalues and eigenvectors of for Case 1 are depicted in Equations (34) and (35), respectively:

Considering the validity of the eigenvectors in Equation (35), along each characteristic line, solution of the Riemann problem (Equation (36)) is obtained.

Using the trapezoidal rule, as in Equation (37), the velocity of the constant state is calculated by Equation (38).

After the determination of and, the vector variable is determined for each inter face. The new variable considering the second hyperbolic part is then calculated by applying the Riemann invariants principle [19,20]:

The following formulation (Equation (40)) is obtained when is integrated over the cell for.

In Case 2, eigenvalues (and) and eigenvectors (and) of are respectively Equations (41) and (42):

As in Case 1, the Riemann invariant is expressed by Equation (43).

The second part of the solution of the Riemann problem (Equation (43)) is used to calculate the velocity of the constant state (Equation (44) or (45)).

The quantities and yield the vector variable

.

The new variable taking into account the second hyperbolic part is then calculated by applying the principle of Riemann invariants:

Integrating Equation (46) over the cell yields for:

The last cell is calculated by considering either the first component of Equation (37) for or the second component of Equation (44) for. Each of the resulting Equations (48) or (49) is combined with the known flow condition:

with dependant respectively on and.

The source term is solved by Equation (50) to estimate the variable vector of each cell depending on, as presented by Guinot [

where indicates that is used to evaluate the source term of the Equation (13).

After determining the variables and, velocity, density, pressurewave celerity by Equation (24), and pressure by Equation (25) are determined for each cell.

Two case studies ((1) a closed downstream valve and (2) a closed upstream valve) are performed, and numerical results compared to the experimental results.

The used experimental results are from the study of Adamkowski and Lewandowski [^{2}/s. The pressure wave celerity calculated according to the pipe characteristics is m/s. The initial velocity in the pipe before closing of the valve is.

Comparison of numerical results with measured results needs to consider several parameters such as the rate of air (Equation (24)) and the static friction. As shown by Wylie and Streeter [

Comparison between simulated and measured results (

The experimental tests used to examine the unsteadyflow, in case of a closed valve installed upstream just after the pump are taken from Pezzinga and Scandura [^{11} N/m^{2}, roughness 0.1 mm) fed by a centrifugal electropump. A 1 m^{3} pressure tank is located at the downstream end of the pipe. Closing the upstream valve generates an interesting transient flow useful for analyzing dynamic friction.

The pressure variation is measured by pressure transducers located at the upstream valve and at the middle of the pipe. The average temperature of the water during the tests was 15˚C; the values of the kinematic viscosity and of the elastic modulus K were assumed to be 1.14 × 10^{−6} m^{2}/s and 2.14 × 10^{9} N/m^{2}, respectively. The theoretical pressure wave celerity considered by the authors [

component enables a better agreement between calculated and measured pressure values. The maximum difference obtained is 6%, and is greatest at the valve position. This difference may be due to the boundary condition calculation. The difference between measured and calculated pressure values is very low in the first pressure peaks, i.e., just after the valve closure. These pressure peaks are the most damaging to water plants. Overall, the numerical results tend to overestimate pressure values, especially at the valve position, which may be due to the choice of static friction, the initial flow condition before the valve closure, the boundary condition calculation, or the early stage of computing. Another factor that might affect the quality of results is the choice of the rate of air. The choice of the polytropic coefficient may also influence the quality of results. Experiments are rarely realized under ideal adiabatic or isothermal conditions. Thus, tests with a uniform measure of the rate of air are needed to better calibrate the model.

This paper focuses on the resolution of unsteady friction using the Godunov approach in a finite volume method with single-equivalent two-phase flow equations. The calculated results show good agreement with experimental measurements, especially for the first pressure peaks, which are the most dangerous for pipe safety. However, dissipation is found to be lower in the calculated than in the measured results. The differences between simulated results with a dynamic friction component and those with a static friction component are more important in the case of a closed upstream valve than a closed downstream valve. It would appear that taking the dynamic friction component into account is more relevant in the case of upstream valve closure. This could be due partly to the formalization of the dynamic friction component, and in particular, the inclusion of acceleration and deceleration, as well as direction of flow. The difference could also be due to the calculation of boundary conditions, which are different in the two case studies. These results demonstrate that the proposed approach allows dynamic friction to be taken into account in finite-volume models, using the increasingly popular Godunov approach. The results also point to the possibility of considering the effect of air in order to improve the quality of simulation models. The model requires further improvements to more accurately reproduce the pressure values in cases of upstream valve closure. Searching for an adequate solution should address the determination of boundary conditions, the formalization of the dynamic friction component and the choice of the polytropic coefficient and experiments in which all parameters are adequately defined.