3. Numerical schemes

In any gas turbine, the flow of the fluid in the compressor is always more unsteady and turbulent, which further makes the design more complicated. For all turbulent flows, the governing equations are the unsteady Navier-Stokes (N-S) equations. But those equations are very much difficult to solve. The following are the governing equations in tensor notation for understanding the basic nature of the equations as presented by [4].

$$\frac{\partial \rho u\_i}{\partial t} + \frac{\partial \left(\rho u\_i u\_j\right)}{\partial x\_j} = -\frac{\partial (p)}{\partial x\_i} + \frac{\partial}{\partial x\_j} \left(\mu \left[\frac{\partial (u\_i)}{\partial x\_j} + \frac{\partial (u\_j)}{\partial x\_i}\right]\right) - \frac{\partial}{\partial x\_i} \left(\frac{2}{3}\mu \frac{\partial (u\_k)}{\partial x\_k}\right) + S\_l \tag{16}$$

Energy equation is given by

$$\frac{\partial \rho \epsilon}{\partial t} + \frac{\partial (\rho u\_{\dot{\gamma}} \epsilon)}{\partial x\_{\dot{\gamma}}} = -p \frac{\partial (u\_{\dot{\gamma}})}{\partial x\_{\dot{\gamma}}} + \frac{\partial}{\partial x\_{\dot{\gamma}}} \left( k \frac{\partial (T)}{\partial x\_{\dot{\gamma}}} \right) + \mathcal{Q} + S\_{\dot{\imath}} \tag{17}$$

The problems of physical engineering, which involve high turbulence when modeled based on RANS equations, are also called statistical turbulence models because the method strongly involves statistical averaging procedure. Consequently, the computational effort to solve

Numerical Simulations of a High-Resolution RANS-FVDM Scheme for the Design of a Gas Turbine Centrifugal…

As discussed in the earlier articles, due to the involvement of more unknown parameters in the modeling of turbulent problems, the accuracy of the model becomes very much difficult and leads to erroneous results. Hence, it is required to develop a numerical procedure to close or converge the system of equations. One such two equations category turbulence model is

In this model, the velocity of the turbulent flow and the length scales are independently calculated using two different equations. The basic model is the standard k-epsilon model [4].

The velocity of the turbulence can be calculated by forming a model for the corresponding

The length scale is represented by ε, which is the rate of dissipation and can be calculated from

∂xj

where Pk represents the turbulence kinetic energy due to the mean velocity gradients, Pb represents the turbulence kinetic energy due to buoyancy, and YM represents the contribution of the

The difficulty in the design of the compressor is because it involves two vital phases of the design. One being the 1D design of the compressor and the other is the deep numerical analysis of the design. The difficulty in the first phase is overcome by using mean line theory. In principle, the mean line theory follows the preliminary design that is carried out by neglecting the air flow variations in radial direction and the location of the mean blade radius

∂xj

þ C<sup>1</sup><sup>E</sup> E k

þ Pk þ Pb � rε � YM þ Sk

ð Þ� Pk þ C<sup>3</sup>EPb C<sup>2</sup>Er

ε2 <sup>k</sup> <sup>þ</sup> <sup>S</sup><sup>E</sup>

http://dx.doi.org/10.5772/intechopen.72098

103

(20)

(21)

<sup>μ</sup> <sup>þ</sup> <sup>μ</sup><sup>t</sup> σk <sup>∂</sup>ð Þ<sup>k</sup>

RANS equations is very less when compared to the other schemes.

3.2. Governing equations for turbulence models

K-epsilon model [4].

3.2.1. K-epsilon models

3.2.1.1. Standard k-ε model

the Eq. (4)

kinetic energy using the following Eq. (4)

∂ð Þ rkui ∂xi

> ¼ ∂ ∂xj

∂ð Þ rEui ∂xi

¼ ∂ ∂xj

<sup>μ</sup> <sup>þ</sup> <sup>μ</sup><sup>t</sup> σE <sup>∂</sup>ð Þ<sup>E</sup>

fluctuating dilatation in compressible turbulence to the overall dissipation rate.

∂ð Þ rk ∂t þ

∂ð Þ rE ∂t þ

3.3. Mean: Line analysis

is considered for the analysis.

where

$$\mathcal{Q} = \left( \mu \frac{\partial(u\_i)}{\partial \mathbf{x}\_j} \left[ \frac{\partial(u\_i)}{\partial \mathbf{x}\_j} + \frac{\partial(u\_j)}{\partial \mathbf{x}\_i} \right] \right) - \left( \frac{2}{3} \mu \left( \frac{\partial(u\_k)}{\partial \mathbf{x}\_k} \right)^2 \right) \tag{18}$$

The main disadvantage in solving N-S equations is computing. This is due to the reason that to represent even very small scale of velocity and pressure fluctuations using N-S equations, temporal resolution is required as the N-S equations are spatial fineness equations. Furthermore, the accuracy and resolution of the scheme reduce due to the increase in accumulative rounding off value errors because of the increase in grid points to achieve fine meshes.

Hence, more accurate schemes are required to achieve the solutions of turbulence models with highest resolution. One such numerical scheme, which can effectively solve the turbulence problems, is Reynolds-averaged Navier-Stokes (RANS) equations.

### 3.1. Reynolds-averaged Navier-Stokes (RANS) equations

The transport equations should be modified by introducing the components with averaged and fluctuating components for solving the turbulent models. RANS are the equations of the motion of the fluid flow with time-averaged equations. If flow turbulence properties are known, then suitable approximations can be made and high-resolution solutions to the N-S equations can be achieved by solving the RANS equations. The RANS equations in tensor notation are described below [4]

$$\rho \frac{\partial \overline{u\_i}}{\partial t} + \rho \frac{\partial (\overline{u\_i u\_j})}{\partial x\_j} = \rho \overline{f\_i} + \frac{\partial}{\partial x\_j} \left( -\overline{p} \delta\_{ij} + 2\mu \overline{S\_{ij}} - \rho \overline{u\_i' u\_j'} \right) \tag{19}$$

where Sij <sup>¼</sup> <sup>∂</sup>ð Þ ui ∂xj þ <sup>∂</sup>ð Þ uj <sup>∂</sup>xi is the mean rate of strain tensor. The problems of physical engineering, which involve high turbulence when modeled based on RANS equations, are also called statistical turbulence models because the method strongly involves statistical averaging procedure. Consequently, the computational effort to solve RANS equations is very less when compared to the other schemes.

### 3.2. Governing equations for turbulence models

As discussed in the earlier articles, due to the involvement of more unknown parameters in the modeling of turbulent problems, the accuracy of the model becomes very much difficult and leads to erroneous results. Hence, it is required to develop a numerical procedure to close or converge the system of equations. One such two equations category turbulence model is K-epsilon model [4].

### 3.2.1. K-epsilon models

3. Numerical schemes

102 Numerical Simulations in Engineering and Science

∂rui ∂t þ

Energy equation is given by

where

nature of the equations as presented by [4].

∂ ruiuj � � ∂xj

> ∂rE ∂t þ

∅ ¼ μ

¼ � <sup>∂</sup>ð Þ<sup>p</sup> ∂xi þ ∂ ∂xj μ ∂ð Þ ui ∂xj þ ∂ uj � � ∂xi

∂ ruje � � ∂xj

> ∂ð Þ ui ∂xj

problems, is Reynolds-averaged Navier-Stokes (RANS) equations.

∂ uiuj � � ∂xj

¼ rf <sup>i</sup> þ

<sup>∂</sup>xi is the mean rate of strain tensor.

3.1. Reynolds-averaged Navier-Stokes (RANS) equations

notation are described below [4]

∂xj þ <sup>∂</sup>ð Þ uj

where Sij <sup>¼</sup> <sup>∂</sup>ð Þ ui

r ∂ui <sup>∂</sup><sup>t</sup> <sup>þ</sup> <sup>r</sup> ¼ �p

∂ð Þ ui ∂xj þ ∂ uj � � ∂xi

� � � �

∂ uj � � ∂xj þ ∂ ∂xj k ∂ð Þ T ∂xj � �

The main disadvantage in solving N-S equations is computing. This is due to the reason that to represent even very small scale of velocity and pressure fluctuations using N-S equations, temporal resolution is required as the N-S equations are spatial fineness equations. Furthermore, the accuracy and resolution of the scheme reduce due to the increase in accumulative

Hence, more accurate schemes are required to achieve the solutions of turbulence models with highest resolution. One such numerical scheme, which can effectively solve the turbulence

The transport equations should be modified by introducing the components with averaged and fluctuating components for solving the turbulent models. RANS are the equations of the motion of the fluid flow with time-averaged equations. If flow turbulence properties are known, then suitable approximations can be made and high-resolution solutions to the N-S equations can be achieved by solving the RANS equations. The RANS equations in tensor

> ∂ ∂xj

�pδij þ 2μSij � ru<sup>0</sup>

� �

i u0 j

rounding off value errors because of the increase in grid points to achieve fine meshes.

In any gas turbine, the flow of the fluid in the compressor is always more unsteady and turbulent, which further makes the design more complicated. For all turbulent flows, the governing equations are the unsteady Navier-Stokes (N-S) equations. But those equations are very much difficult to solve. The following are the governing equations in tensor notation for understanding the basic

� � � �

� <sup>2</sup> 3 μ � ∂ ∂xi

∂ð Þ uk ∂xk � �2 !

2 3 μ ∂ð Þ uk ∂xk � �

þ Si (16)

(18)

(19)

þ ∅ þ Si (17)

In this model, the velocity of the turbulent flow and the length scales are independently calculated using two different equations. The basic model is the standard k-epsilon model [4].
