# Turbulent Models On Boundary Layer Transition Engineering Essay

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The present study reports on a Shear Stress Transport (SST) RANS linear eddy viscosity based turbulent models on boundary layer transition through a linear turbine cascade. A number of analyses are made with results from earlier direct numerical simulations of grid turbulence through the similar cascade. The results of numerical simulation of two-dimensional turbulent flow on low pressure turbine T106A cascade are presented. Employing turbulence model, k-w model, a numerical and predictability analyses was done on computational fluid dynamics in comparison with previous direct numerical simulations done by Wissink and Durbin (2003). It was been concluded that the fine mesh and computational grids are required to get effective data of the description of the turbulent flow condition and that SST approach at certain extent can be feasible as an analyses tool for flow separation as compared to direct numerical simulation (DNS). The results achieved in this study have shown the difference in behaviour of the air flow, when varying the incoming flow angle, such as the difference in pressure, velocity and the turbulent kinetic energy that is occurring in the flow.

## Nomenclature

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RANS

DNS

CFD

LPT

HPT

LPT

## Motivation

The study is entitled "CFD analyses in the low pressure T106A turbine cascade geometry". The T106 blade is a high lift turbine blade used for research purposes. A high lift blade is designed to enhanced loading and offers lower blade counts by extracting the maximum capacity of power per blade. However a small number of blades are necessary for power extraction compared to the normal blades hence reducing the amount of the weight. The deceleration of the flow over part of suction side of the blade provides increase to an adverse pressure gradient on the suction side of the blade. The adverse pressure gradient affects boundary layer development and cause the flow separation. The separation flow leads to decrease of kinetic energy and total relative pressure at early stage. The LPT efficiency and fuel consumption is greatly affected by these adverse occurrences and also affects the aerodynamics of the blade. DNS method is being used for the study of upstream wakes and turbulence at low Reynolds number at an angle of attack (AoA) of 45.5 and 37.7 degrees in the lack of free stream turbulence by Wissink and Durbin respectively. As Shear Stress Transport turbulence (SST) modelling is less costly compared to the DNS, an effective outcome in predicting the flow behaviour after comparing with DNS will validate the use of SST as method and tool for modelling and predicting flow separation.

## Aim

The purpose of this study is to use CFD numerical methods to simulate and analyse turbulence flow through a low pressure T106A turbine cascade and the analyses of the effect of free stream turbulence intensity on boundary layer separation. Be able to compare the Shear Stress Transport turbulence model with results obtained in the DNS.

## Introduction

Computational Fluid Dynamics (CFD) uses numerical and algorithms to simulate and analyze the problems that involve fluids flow. The field of CFD simulations is strongly dependent on the evolution of the computer related innovation and technology and on the user understanding and solving partial and ordinary differential equations. However, numerical solving of complex flows in real existing conditions requires overpowering computational capacity in solving such problems, which is more dependent on the physical models applied and understanding of physical phenomena that are dominant at certain conditions. Modelling of flow and in particular turbulent flow is a complex activity, time dependent and has random motions is one the reasons for the aforementioned complexity. Up to date there is no model or theory which can effectively capture the effects of existing types of turbulent problems under all conditions.

There are practical and several approaches being carry out for modelling and simulation of turbulent flow, however the selection for the aforementioned conditions depends on the cost of running the simulation, effective solution, level of flow problem description, accuracy, range and applicability. Each model has its unique condition of flow and may depend on the assumptions made and the nature of the problem to be investigated. The existing simulation techniques, such as Direct Numerical Simulation (DNS) yields in terms of instantaneous velocity while solving equations of fluid flow and highly accurate simulation technique, being free form flow dependent parameters it is considered to be effective simulation method compared to other methods, such as Reynolds Averaged Navier-Stokes (RANS) which solves the average velocity and less expensive to run than DNS. The accurateness of the simulation methods and model can be analysed by comparison of data that can be from experimental or other type of simulation method. For the present study a comparative analyses is made between the SST k-omega model (Linear eddy viscosity models) and DNS method on low pressure turbine T106A cascade.

## Literature review

## Gas Turbine Engine

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The majority of the aircrafts in service today generate their propulsive power by use of gas turbine engines. The atmospheric air go into engine is drawn in by a fan and subsequently compressed by several rows of axial compressor blades thus generating enough stagnation pressure to generate thrust. The compressor has two sections. In the case of 2-spool engine, the low pressure compressor (LPC) immediately follows the fan and high pressure compressor (HPC) succeeds the low pressure compressor and leads to the combustor. On the other hand, high by-pass ratio fan engine feature an intermediate pressure system, known as triple-spool system. Where, each section of the compressor has its own row of blades or stages which pressurize the incoming air. This pressurized air is mixed with fuel and ignited to produce a high energy flow which exits into the turbine section. The turbine section extract energy from the hot gases incoming from the combustor, and drives the compressor stages. The high pressure turbine (HPT) is placed after the combustor and drives the HPC, whilst the low pressure turbine (LPT) which is situated after the HPC drives the low pressure compressor (LPC). The HP turbine blade operates in very hostile environment with temperatures ranging from 800 oC - 1700 oC [4]. (Rolls Royce - The Jet Engine)

Figure 1 - Diagram of a high-bypass turbofan engine

Retrieved from internet: http://en.wikipedia.org/wiki/File:Turbofan_operation.svg [accessed on December 2010]

In order to sustain the blades from melting as result of high temperature conditions, the turbine blade geometry has a cooling system mechanism in the form of internal passages which circulate cooling air and dump the heat through a microscopic hole on the blade surface to cool the blades. For current study, the focus is down to the low pressure turbine (LPT). The main function of the LPT is to drive the low pressure compressor (LPC) by extracting the incoming energy from the expanding gases which are released from HPT. The drift in modern low pressure turbine is to drive larger fans at lower speeds and at the same time reduce the complexity and weight [4]. The LPT section compromises of several stages, with each stage employing one row of stationary nozzle guide vane and moving blades. The number stages is determined by the relationship between power required from the gas flow, the rotational speed at which power needs to be generated and the diameter of the turbine required[5]. Then nozzle guide vanes (NGV) directs the incoming flow in the tangential directional of rotation and by coalescent action of impulse and reaction on the LPT blade by the flow discharged by the NGV's, rotation of the shaft is attained.

The aircraft engine low pressure turbine operates at extreme conditions, whereas at the take-off conditions the LPT operates under a high Reynolds number and maximum loading condition at sea level. Whilst at the cruise conditions it operates at low density, hence the low Reynolds number. This condition causes low momentum flow and makes the LPT venerable to boundary layer separation, which are cause by the existence of an adverse pressure gradient on the suction side of the LPT blade. The aforementioned condition is difficult to predict and are not fully accounted for during the LPT design, and thus the gradual loss in performance of the LPT. The flow in low pressure turbine section is affected by two aspects.

First, the affect of the Reynolds number and the second, the periodic disturbance caused by the upstream row of blades that produces wakes that is passively passed to downstream by the main flow. Due to the low Reynolds number, the boundary layer along the LPT will initially be laminar. A laminar boundary layer is more prone to separation when subjected to adverse pressure gradient than a turbulent layer. The separation of flow results in a low pressure separated region between the blade wall and the main flow, this affects the efficiency of the LPT which strongly affects the engine thrust and a decrease of the fuel efficiency.

## Research journals

To gain understanding of all the tasks and simulations required for the project, an in depth literature review was carried out. Durbin [1], conducts is research in T106A turbine cascade for an incidence of angle of 37.7o, reports of direct numerical simulation of turbulence model through a linear turbine blade cascade in order to investigate pattern of turbulent kinetic energy generated by external effects and distortion on the boundary layer interaction. Distorted kinetic energy has been found to rise inside the passage in the direction of the trailing edge of the pressure surface and at stagnation regions. This rise has been to some extent attributed to the convection process of inflow turbulence. For turbulence free inlet, the natural transition takes place alongside the suction side of the trailing edge. For grid turbulence and wake inlets, bypass transition occurs further upstream triggered by the convection of the inlet disturbances of the boundary layer of the blade.

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Examples of our workFigure 2 - Computational domain for blade passage DNS with grid line plotted [1].

Wissink [2], conducts the research on three dimensional DNS, Direct Numerical Simulations. The grid simulation is carried out in low pressure turbine cascade flow at high angle of attack of 45.5 o and at Reynolds number of 51,831. The selected angle of attack was found to cause flow separation at the leading edge and in location upstream of the trailing edge on the suction side of the blade. It was observed that the boundary layer on the downstream half of the suction side tends to separate as soon as the free stream turbulence of the passing wakes has moved downstream. During the presence of the free stream turbulence vortices vanishes, however appears again when the free disturbances have passed.

High amounts of fluctuating kinetic energy are generated in the re-circulating flow raised by the disturbances in the flow itself. As they move downstream, the re-circulating rolls gradually tend to vanish. Also, elongated vertical structures were observed on the upstream half of the suction side which is result of the elongation of the wake vortices by the local flow acceleration.

## The Navier-Stokes Equation

Navier-Stokes equations are set to describe the motion of fluid flow, these equations are derived by applying Newton's second law to fluid motion, and set the changes in momentum of the particles of a fluid, and these are simply the product of pressure gradient and dissipative viscous forces acting inside the fluid. The main assumptions used to derive the Navier-Stokes equations are as follows [4]:

Viscous fluid model constants such as density and viscosity require experimental data.

When the fluid is at the rest condition, the stress is hydrostatic.

The fluid motion is continuous, homogeneous, and isotropic.

When the flow is in pure dilation, average stress is equal to the pressure.

These assumptions do not exclude turbulent flow; hence the Navier-Stoke equations can be used for describing turbulent flows as long as they do not flout the above assumptions. According to law of conservation of momentum:

Rate of momentum accumulation = Rate of momentum in - Rate of momentum out + Sum of forces acting the system

The mathematical form of the above statement can be expressed by using velocity components u, v and w to calculate the terms expressing momentum at very face of the control volume under consideration, and velocity gradients to express the stresses in terms of the fluid viscosity [4].

For an incompressible flow, the Navier Stokes equations can be written as:

## Continuity Equation:

## Momentum equations

Components of velocity: u, v, w

Components of direction: x, y, z

## Energy equation

The terms of left hand side of the momentum equation, whilst on the right hand side pressure gradient representing the diffusion and body forces respectively. Where, is the heat conductivity, T is the temperature and cp is the specific heat. The energy equation is beyond this study, as it is relevant in study of incompressible flow on LPT.

Generally the turbulent flow is dominated by recirculation, eddies, wakes. Solving the Navier-Stokes equations for each scale of motion in turbulent flow requires use of fine meshing around the area of interest. This requires that the computational of flow properties at each node of the fine mesh will require substantial time and large data storing capacities.

Applying mathematical methods such as Reynolds decomposition with Navier-Stokes equation, the mathematical expression generated is defined as Reynolds Average Navier-Stokes (RANS) equation, this combination yields itself as a useful method to compute flow characteristics without giving into large computing requirements.

## Turbulence

Turbulence is a flow regime characterized by chaotic and unpredictable changes in flow properties and includes elements like three dimensional vorticity, and a wide spectrum of scale of motion. The main importance in study of turbulent flows is attributed to the most naturally occurring flows, and a range of technological applications evidences turbulent characteristics. The theory of turbulence in accordance with Von Karman (1937), Prandtl (1925) and Taylor (1935) can be categorized in two approaches:

Study of turbulence leading to the semi empirical theory, which provides data on the direct applications (e.g. skin friction on pipe and airfoils).

Study of turbulence under idealized conditions such as isotropic or homogeneous turbulence.

## Turbulence models

A turbulence models are a computational method to close the system of mean flow equations. Turbulence models allow the calculation of the mean flow without first calculating the full time-dependent flow field [7]. Turbulence models are statistical closure models, consisting of partial differential equations for time mean turbulence parameters to which the unknown correlations can be linked, or equations for the unknown correlations themselves [8]. An alternative to resolve the spatial and time evolution of turbulence on the direct numerical simulation within a turbulent flow is to integrate the governing flow equations in the time and thus derive the equations for the time averaged flow properties. In this approach, all time dependent features are integrated into statistical correlations which arise from time averaging the Navier-Stokes equations [9].

## Classification of turbulent models

Recently, turbulent flows may be computed by applying several different approaches, either by solving the Reynolds Averaged Navier-Stokes equations with suitable models for turbulent quantities. The main approaches as follows:

## Reynolds-Averaged Navier-Stokes (RANS) Models - Linear eddy viscosity models

Zero equation model - Mixing length model.

One equation model - Spalart-Almaras.

Two equation models: k- style models (standard, RNG, realizable), k- model.

The number of equations defines the number of additional partial differential equations that needs to be solved.

## Computation of fluctuating quantities

Large-eddy simulation (LES) - one computes time-unstable flow, however models sub-grid scale motion.

Direct numerical simulation (DNS) - no modeling is applied; one is required to resolve the smallest scales of the flow as well. This means that whole range of spatial and time scales of turbulence must be resolve in the computational mesh.

Figure 3 - Turbulent Models prediction methods [10].

The extend for modelling certain computational fluid flow approach is illustrated above, it can be seen that models DNS and LES resolve shorter length scales than models solving RANS equations. Additionally they have ability to generate better results. However DNS and LES have a demand of high and greater computer power than those models applying RANS methods [10]. The RANS turbulent model was selected for this study as aforementioned and Shear Stress Transport (SST) k-omega model used as part of the simulations for analyses and prediction of flow separation in low pressure turbine cascade.

## Turbulence model used in this study

The Shear Stress Transport (SST) k-omega turbulence model is a two equation eddy viscosity model, which is very well known by solving a turbulence frequency based model (k-omega) at the wall and can be used as low Reynolds number turbulence model without any additional damping role. Whilst, k-epsilon in the bulk flow and avoids the ordinary k-omega problem where the model is too sensitive to the inlet free-stream turbulence properties. The Shear Stress Transport model is considered and rated the best and accurate model for aerodynamic purposes according to NASA.

## Shear Stress Transport (SST) governing equations

The Shear Stress Transport model solves two equations, two scalars, usually the turbulent kinetic energy k, and specific dissipation rate,. [11] The two equation model is given by:

## Turbulence kinetic energy

## Specific Dissipation Rate

And turbulent eddy viscosity is computed from:

## Advantages:

Good behavior in adverse pressure gradients and separating flow than k-epsilon.

Robust, and accurate in ANSYS CFX software.

Predicts accurately the skin friction and velocity profile.

Economical and less memory capacity requirement

## Disadvantages:

Does not give consistent results in wakes and free shear layers.

AoA

Model

Intensity level

Observation

37.70

SST

Low

37.70

SST

Medium

37.70

SST

High

45.50

SST

Low

45.50

SST

Medium

45.50

SST

High

00

SST

Low

00

SST

Medium

00

SST

High

Results and Discussion