Digital Analysis of Common Turbulence Patterns in Centrifugal Pump Flow Simulation Based on COMSOL Multiphysics Software

Centrifugal pumps remain the most popular configuration in the steel industry thermal and nuclear power plants as well as domestic applications. They operate at very high speeds and convey semi-viscous fluids causing flow turbulence. The latter attracting the attention of researchers to examine more thoroughly the solutions of the Navier-Stokes equation in turbomachinery using CFD [1, 2]. CFD in centrifugal pumps was the main focus of several researchers including Hedi et al. [3], Mentzos et al. [4], Shah et al. [5] which predicted the centrifugal pump performance by CFD analysis under various operating conditions.

Furthermore, Bacharoudis et al. [6], Anagnostopoulos [7], Patel and Ramakrishnan [8] conducted parametric CFD simulation studies to numerically examine the impact of varying impeller and volute geometry parameters on pump efficiency, head, and performance. The CFD simulation process necessitates a powerful computing tool, adequate simulation software, and a thorough consideration of the turbulence model to ensure success. Turbulence modeling [9, 10] is one of the three fundamental aspects of Computational Fluid Dynamics (CFD). Sophisticated mathematical theories have been elaborated regarding the two other vital components: grid generation and algorithm development. The k-ε standard, the k-ε low Re, the k-ε RNG, the k-ω standard, and the k-ω SST models are among the well-known turbulence models [10-13].

The computational performance, according to the turbulence models, has been and proved to be quite distinct.

The performance under the k-ε EARSM model performs well against four other models: k-ε, SSG Reynolds Stress, RNG k-ε, k-ω) [14]. In contrast, the k-ω model resulted in more realistic velocity profiles that systematically generated excessively high turbulent shear stress values [15].

Different tests revealed that both the k-ε realizable and SST transition turbulence models provide better outcomes in the supersonic flow calculation, which is typical in advanced jet engines [16]. Besides, the comparison between this model indicates that the k-ε model forecasts the flow inside the centrifugal pump accurately enough [17]. As far as the pump's mean flow field is considered, the SAS model has no advantage over the SST model [18].

The aforementioned reviews confirm that it is essential to develop reliable models to optimize the design of centrifugal pumps and predict their performance. Accordingly, the present research paper concentrates on the evaluation of three turbulence models: k-ε, k-ω, k-ω SST in order to predict the velocity and pressure distributions inside a centrifugal pump as well as the pump performance. A substantial part of this work has been devoted to investigating different turbulence models effect on iteration number, computational time and Reynolds number, g the grid sensitivity is also analyzed using Comsol Multiphysics as this is rarely stressed in most previous research studies.

The remainder of the paper will be organized into five sections: Section 2 depicts the pump characteristics considered in this study, Section 3 is devoted to introducing the numerical model, simulation results are discussed in Section 4; Lastly, Section 5 conclusion.

This section explains the results of the research and offers a comprehensive discussion. The results are displayed in figures, graphs, tables, and others.

3.1 Impeller design

A Centrifugal pump translates rotating energy, usually delivered via a motor, to a moving fluid. The two primary components involved in the energy conversion process are the impeller and the casing. Passing through the impeller, the fluid gains both velocity and pressure [20, 21]. As a result, the impeller itself is a key part of a centrifugal pump, since it acts as the kinetic energy source. The impeller features, which are studied are summarized in Table 2.

Table 2. Impeller main characteristics

2.png

Figure 2. Impeller detailed plan [9]

The impellers can either be open, semi-open, or closed. As a further point, the impellers could be single or double suction. A single-suction impeller enables liquid to flow into the blade center from a single direction. A double-suction impeller will allow liquid to penetrate to the center of the impeller blades from both directions concurrently [20]. A single-suction closed impeller design will be considered in this study. The pump design was carried out using SolidWorks employing the actual dimensions, which are shown in Figure 2.

3.2 Casing design

The pump casing supplies a pressure limit to the pump and includes channels to correctly steer suction and discharge flow [10]. Furthermore, it decelerates the liquid flow. As per Bernoulli's principle, the volute converts kinetic energy into pressure by reducing the water velocity whilst increasing the pressure. The volute casing reviewed here has been designed per the coil pump's dimensions. The resulting assembly geometry is exhibited in Figure 3.

3.png

Figure 3. 3D view of the global final geometry

3.3 Mesh structure

A free tetrahedral mesh was applied for the global final geometry. The mesh characteristics are summarized in Table 3 and the final mesh structure is illustrated in Figure 4.

Table 3. Mesh composition

4.png

Figure 4. Free tetrahedral mesh of the studied geometry

3.4 Frozen rotor study

The Frozen Rotor study was used to compute the velocity, pressure, and turbulence in Comsol Multiphysics. The frozen rotor approximation implies that the flow in the rotation domain, as stated in the rotation coordinate system, is entirely developed. For that reason, it is generally used in rotating machinery and is viewed as a special case of a Stationary study. The rotating parts are kept frozen in position, and the rotation is accounted for by the inclusion of centrifugal and Coriolis forces. This study is especially suited for the flow in rotating machinery where the topology of the geometry does not change with rotation. It is also utilized to compute the initial conditions for time-dependent simulations of the flow in rotating machinery [22].

3.5 Governing equations

3.5.1 The k-ε turbulence model

The k-ε model is one of the widely used turbulence models for industrial applications. This module includes the standard k-ε model. The model introduces two additional transport equations and two dependent variables: the turbulent kinetic energy, k, and the turbulent dissipation rate, ε. The turbulent viscosity musityisis $\mu_{\mathrm{T}}$ is modeled as [12] Eq. (1):

$f_{\beta}=\frac{1+680 \chi_{k}^{2}}{1+400 \chi_{k}^{2}}, \chi_{k} \leq 0, \chi_{k}>0$

$\chi_{k}=\frac{1}{\omega^{3}}\left(\nabla k_{.} \nabla \omega\right)$

          $\mu_{T}=\rho C_{\mu} \frac{k^{2}}{\varepsilon}$                   (1)

where, Cμ is a model constant; k the Kinetic energy [J].

The transport equation for k reads:

$\rho \frac{\partial \mathrm{k}}{\partial \mathrm{t}}+\rho \mathrm{u} \cdot \nabla \mathrm{k}=\nabla \cdot\left(\left(\mu+\frac{\mu_{\mathrm{T}}}{\sigma_{\mathrm{k}}}\right) \nabla \mathrm{k}\right)+\mathrm{pk}-\rho \varepsilon$                       (2)

where, ρ: The turbulent dissipation rate [m²/s³]; μ_T: Turbulent viscosity [m²/s]; ε: The molecular, eddy viscosity [Pa.s]; P: The Pressure [Pa]; σ_k: Experimental model constant; $\nabla k$: Kinetic energy gradient [J].

The production term Pk is:

$P k=\mu_{T}\left(\nabla u:\left(\nabla u+\nabla u^{T}\right)-\frac{2}{3}(\nabla \cdot u) 2\right)-\frac{2}{3} \rho k \nabla \cdot u$           (3)

where, $\nabla u^{\mathrm{T}}$: Turbulent velocity gradient [m/s]; u: x velocity component [m/s].

The transport equation for $\varepsilon$ reads:

$\rho \frac{\partial \varepsilon}{\partial t}+\rho u . \nabla \varepsilon=\nabla .\left(\left(\mu+\frac{\mu_{T}}{\sigma_{\varepsilon}}\right) \nabla \varepsilon\right)+C_{\varepsilon l} \frac{\varepsilon}{k} p_{k}-C_{\varepsilon 2} \rho \frac{\varepsilon^{2}}{k}$           (4)

where, $C_{\varepsilon l}$: model constants.

The model constants in Eq. (1), Eq. (2), and Eq. (4) are determined from the experimental data [1] and the values are listed in Table 4 [22].

Table 4. Model constants

The constants are: c_μ: k-ε Model constant; c_ε1: Experimental model constant; c_ε2: Experimental model constant; σ_k: Dissipation per unit turbulent kinetic energy [m²/s²]; σ_ε: Closure coefficients in turbulence kinetic-energy equation.

3.5.2 The k-ω turbulence model

The k-ω model solves for the turbulent kinetic energy, k, and the dissipation per unit of turbulent kinetic energy, ω. The CFD Module has the Wilcox revised k-ω model [22, 23]:

$\rho \frac{\partial k}{\partial t}+\rho u . \nabla k=p_{k}-\rho \beta^{*} k \omega+\nabla \cdot\left(\left(\mu+\sigma^{*} \mu_{T}\right) \nabla k\right)$                     (5)

$\rho \frac{\partial \omega}{\partial t}+\rho u . \nabla \omega=\alpha \frac{\omega}{k} p_{k}-\rho \beta^{*} \omega^{2}+\nabla .\left(\left(\mu+\sigma \mu_{T}\right) \nabla \omega\right)$                   (6)

where, α: Model constant; ω: Closure coefficients in the specific dissipation rate equation; β^*, σ^*: Turbulence-model coefficient; t: Time [s]; $\mu_{\mathrm{T}}=\rho \frac{\mathrm{k}}{\omega} ; \alpha=\frac{13}{25} ; \beta=\beta_{0} f_{\beta} ; \beta^{*}=\beta_{0}^{*} f_{\beta} ; \sigma=\frac{1}{2} ; \sigma^{*}=\frac{1}{2} ; \beta_{0}=\frac{13}{125}$; $f_{\beta}=\frac{1+70 \chi_{\omega}}{1+80 \chi_{\omega}} ; \chi_{\omega}=\left|\frac{\Omega_{i j} \Omega_{j k} \Omega_{k i}}{\left(\beta_{0}^{*} \omega\right)^{3}}\right| ; \beta_{0}^{*}=\frac{9}{100} ; f_{\beta}:$ Round-jet function.

Ω_ij is the mean rotation-rate tensor:

$\Omega_{i j}=\frac{1}{2}\left(\frac{\partial \overline{\mathrm{u}}_{i}}{\partial \chi_{j}}-\frac{\partial \overline{\mathrm{u}}_{j}}{\partial \chi_{i}}\right)$                  (7)

S_ij is the mean strain-rate tensor:

$\mathrm{S}_{i j}=\frac{1}{2}\left(\frac{\partial \overline{\mathrm{u}}_{i}}{\partial \chi_{j}}+\frac{\partial \overline{\mathrm{u}}_{j}}{\partial \chi_{i}}\right)$                  (8)

Pk is given by Eq. (3). The following auxiliary relations for the dissipation, ε, and the turbulent mixing length, l∗, are also used:

$\varepsilon=\beta^{*} \omega \mathrm{k} ; l_{\text {mix }}=\frac{\sqrt{k}}{\omega}$                  (9)

3.5.3 The k-ω SST turbulence model

To combine the superior behavior of the k-ω model in the near-wall region with the robustness of the k-ε model, Menter [24] introduced the SST (Shear Stress Transport) model which interpolates between the two models. The version of the SST model in the CFD module includes a few well-tested [25] modifications, such as production limiters for both k and ω, the use of S instead of Ω in the limiter for μ_T and a sharper cut-off for the cross-diffusion term. It is also a low Reynolds number model. In other terms, it does not apply wall functions. The Low Reynolds number refers to the region, which is close to the wall where viscous effects dominate. The model equations are formulated in terms k and ω, [22]:

$\rho \frac{\partial k}{\partial t}+\rho$ u. $\nabla k=p-\rho \beta_{0}^{*} k \omega+\nabla \cdot\left(\left(\mu+\sigma_{k} \mu_{T}\right) \nabla k\right)$

$\rho \frac{\partial \omega}{\partial \mathrm{t}}+\rho u . \nabla \omega=\frac{\rho \gamma}{\mu_{\mathrm{T}}} \mathrm{p}-\rho \beta \omega^{2}+\nabla \cdot\left(\left(\mu+\sigma_{\omega} \mu_{\mathrm{T}}\right) \nabla \omega\right)+2(1-\left.\mathrm{f}_{\mathrm{v} 1}\right) \frac{\rho \sigma_{\omega 2}}{\omega} \nabla \omega . \nabla \mathrm{k}$             (10)

where,

$P=\min \left(P_{k}, 10 \rho \beta_{0}^{*} k \omega\right)$        (11)

And P_k is given in Eq. (3). The turbulent viscosity is given by:

$\mu_{\mathrm{T}}=\frac{\rho a_{1}}{\max \left(a_{1} \omega, S f_{v_{2}}\right)}$            (12)

where, f_v1 and f_v2 are the auxiliary function in turbulence model, f_v1 is equal to zero away from the surface (k-ε model), and switches over to one inside the boundary layer (k-ω model) [23]. This reflects the efficiency of combining the two models, S is the characteristic magnitude of the mean velocity gradients, a₁: model canstant (0, 31).

$\mathrm{S}=\sqrt{2 \mathrm{~S}_{i j} \mathrm{~S}_{i j}}$                 (13)

The model constants are defined through interpolation of appropriate inner and outer values:

$\phi=f_{v 1} \phi_{1}+\left(1-f_{v 1}\right) \phi_{2}$ for $\phi=\beta, \gamma, \sigma_{k}, \sigma_{\omega}$               (14)

where, the constants $\varphi$ of the model are calculated from the constants $\varphi 1$ and $\varphi 2 ; \gamma$ : Momentum thickness [m].

The interpolation functions f_v1 and f_v2 are defined as:

$f_{v 1}=\tanh \left(\theta_{1}^{4}\right)$

$\theta_{1}=\min \left[\max \left(\frac{\sqrt{k}}{\beta_{0}^{*} \omega l_{w}}, \frac{500 \mu}{\rho \omega l_{w}^{2}}\right), \frac{4 \rho \mu \sigma_{\omega 2} k}{C D_{\mathrm{k} \omega} l_{w}^{2}}\right]$                    (15)

where,

$C D_{\mathrm{k} \omega}=\max \left(\frac{2 \rho \sigma_{\omega 2}}{\omega} \nabla \omega . \nabla \mathrm{k}, 10^{-10}\right)$                 (16)

where, l_w is the distance to the closest wall, and CD_kω positive portion of the cross-diffusion in ω-transport equation.

The main aim of this research was to investigate how three numerical turbulence models could affect both the convergence and the performance of the flow simulation through a comparative computational analysis, which is based on COMSOL Multiphysics, in which the flow modeling process was based mostly upon stationary Navier-Stokes equations. equally, the impact of these models on the numerical CFD simulation was also taken into account.

The analyzed findings allowed us to conclude that the best computational accuracy is obtained by the k-ω model, whereas the lowest was given by the $k-\omega$ SST model. Nevertheless, a significantly low calculation cost was attained through the latter. Likewise, a better pumping performance was also recorded. The collected data of this investigation also evinced the computation under $k-\omega$ SST was successfully able to achieve convergence after only 312 iterations. However, the $k-\omega$ model and the $\mathrm{k}-\varepsilon$ model provided more accuracy at the expense of the computational cost and the memory used.

The proposed study can be implemented in centrifugal pump modeling and design. It is also applicable in:

- Research, which is industrially implemented for centrifugal pump design optimization;

- Further experiments involving COMSOL Multiphysics software for enhancing centrifugal pump efficiencies.