# Turbulence across a partially closed large diameter, square knife gate valve.  A CFD Study

## 1. INTRODUCTION

### 1.1 Case Study

This paper discusses the effect of turbulence in partially closed square knife gate valves on the downstream flow of large diameter pipes.

A series of flow measurements using ultrasonic flowmeters were carried out in two different locations on
a water trunk main with a partially closed valve. The downstream location was equipped with two
panametric
systems in which the data collected from each one was later compared. Furthermore, the chamber upstream was equipped with mini-sonic metering devices comprised of two beams that provided
two sets of data for comparison. The differences in the flow measurements with the two panametric
systems were higher than the differences of the same type of data with the mini-sonic metering devices, hence, this observation could be explained by a swirl occurring on the downstream side of the gate valve.
Another Aqua probe flow meter was installed on the pipe center-line to study the variation of average velocities every minute.

The fluctuations were as high as 7 % indicating the existence of a highly
turbulent flow accompanied by large vortices.

## 2. Theory

### 2.1.1 Navier Stokes equation

In computational fluid dynamics, a fluid domain is divided into finitely small control volumes governed
by a set of partial differential equations. The continuity equation obeys the law of conservation of mass.
A compressible unsteady state flow is conserved:
Where u1, u2, and u3 are the velocities in x1, x2 and x3 directions and ρ is the density of the fluid.
In the case of an incompressible fluid like water, the density is constant with respect to time:

The second equation obeys the conservation of momentum based on the second law of Newton:

is the viscous stress tensor with i and j equal to 1, 2 and 3 for 3D models. Fi is a
body force such as gravitational force. From the momentum and the continuity equations, the Navier
Stokes equations are derived.

Navier Stokes equation:

µ is the molecular viscosity and it is part of the viscous stress equation formulated
by Stokes hypothesis.

### 2.1.2 RANS – Reynolds Average Navier Stokes Equation

It is possible to identify a turbulent flow by studying the stochastic behaviour of the velocity field. Moreover, velocity in the RANS method is divided into a steady mean element and a fluctuating element [2] The instantaneous velocity is equal to:

+

The instantaneous velocities and pressures in the continuity  and momentum equations are replaced by their decomposed values and then time averaged:

is called the Reynolds stress tensor. Using the Eddy viscosity hypothesis this term can be
modeled.
Boussinesq’s famous relation between the Reynolds stress tensor and the mean strain tensor produces the
following equation for Incompressible flows:

υ is the kinematic viscosity which is equal to:

µt is the turbulence eddy viscosity, k is the turbulent kinetic energy and  is the Kronecker delta.[4]
The K-ε model is widely used in industry, it is simple, well tested and satisfies the needs of many
industrial requirements in a timely manner. The model has two transport equations, the first is designated
for the turbulent kinetic energy and the second is for its dissipation ε

The transport equation for K:

I: the rate of change of k with respect to time.
II: transport by convection for k.
III: transport by diffusion for k.
IV: the rate of k production.
V: the rate of k destruction.

The transport equation for ε:

I: the rate of change of k with respect to time.
II: transport by convection for ε.
III: transport by diffusion for ε.
IV: the rate of ε production.
V: the rate of ε destruction.
[5] S is the strain rate.

### 2.2 Large Eddy Simulation – LES

A Large eddy simulation is a CFD model that treats turbulence by resolving the largest eddies and modeling the smaller ones. It applies a low pass filter on the continuity and momentum equations leading to a
filtered Navier Stokes equations with no small scales included in the solution.

Filtered Navier Stokes equation:

The instantaneous velocity is divided into a filtered value and an unresolved value:

The sub-grid scale stresses: appear in the equation due to the filtering process. Using Boussinesq’s hypothesis, the sub-grid stress is evaluated through a relationship with the rate of strain of the resolved flow.
The adopted model in this paper is the Smagorinsky subgrid-scale, it is the simplest model in the LES
family and the most time efficient.

The subgrid-scale turbulent viscosity is modeled by Smagorinsky-Lilly model:

is a function of the strain rate tensor, is the mixing length for subgrid scales which depends
on the cell volume and the distance to the nearest wall.
Smagorinsky’s model requires a near-wall treatment to capture the boundary wall region, hence, damping functions are used to solve this problem and reduce the eddy viscosity near the wall.

Fluent has an incorporated wall treatment function that deals with the
turbulent viscosity at the wall. The law of the wall function is automatically used in Fluent ANSYS in
cases of coarse meshes by relating the mean velocity parallel to the wall to the eddy viscosity in a
logarithmic relationship as follows: