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## COMPUTATIONAL FLUID DYNAMICS

**Finite element method**

Finite element method (FEM) was first introduced in the field of structural mechanics and was later adopted for solving fluid flow problems especially when complex geometry was involved. The ability of the method to decompose the domain into smaller elements or finite elements of flexible geometry highlights the main advantage of the FEM. In addition, in FEM, the weak form of the governing equation is approximated unlike FDM/FVM where we work with the strong form of the governing equations. This allows the ability to provide higher order of approximations as well as flexibility in the implementation of boundary conditions. The weak form is obtained by the inner product of the residual of the governing equations and chosen test functions/basis functions. Such an equation represents the orthogonal projection of the residual error onto the test functional subspace integrated over the domain. Equating this to zero minimises the error functional thus providing the best approximation of the solution to the governing equation. While FEM are eﬀective for structural problems, in the field of CFD involving a large number of grid points, the use of simpler schemes with a lower order approximation such as FDM/FVM is preferred. FEM within CFD is generally applied in the case of complex geometries mainly with industrial applications. For more information about diﬀerent discretisation techniques with examples, please refer to Chung (2010); Pulliam and Zingg (2014).

**Boundary conditions**

Boundary conditions play an important role in CFD. They define the behaviour of the fluid at the boundaries outside of which no information of the fluid flow is known. Thus these conditions need to be defined accurately in space and for transient cases, an additional condition defining the flow at the initial time step is required. An error in the boundary condition can lead to errors in simulation as well as cause issues in convergence and stability of the simulation. In general, the boundary condition can take one of three forms: for a Dirichlet condition, a value for the function is defined at the boundary, while for a Neumann condition, the normal derivative of the function at the boundary is defined and a combination of the two is defined as the Robin boundary condition. Common boundary conditions for CFD (for velocity fields) are:

• Inlet condition defines the value of the fluid flow velocity at the inlet boundary. Wake flows, channel flows, and pipe flows are a few example which use the inlet boundary condition.

• Outlet condition defines the value of the fluid flow velocity at the outlet boundary when this velocity field is known. Channel flows, and pipe flows are a few example which use the outlet boundary condition.

• No slip condition is used to define the velocity at solid boundaries such as the wall of a channel flow. The normal component of the velocity is set to zero while the tangential component is set to the wall speed. The boundary on the cylinder, the walls of a channel or pipe are examples where the no slip condition is implemented.

• Constant pressure condition defines the value of pressure usually at inlet or outlet boundaries especially when the velocity field is unknown. Buoyancy driven flows, free surface flows, and flows around buildings are examples which can employ the constant pressure condition.

• Periodic condition are applied in flows where a repeated flow pattern is observed along a given direction. This condition is applied for the theoretical Taylor-Green vortex simulations, as well as in channel flow for inlet/outlet boundaries and for wake flows in the spanwise direction of the cylinder.

### Turbulence modelling

To accurately simulate a fluid flow, all the length and time scales of the fluid need to be resolved explicitly by the discretisation mechanism employed. However, for the case of turbulent flows, the range of length scales increases exponentially with the Reynolds number and is, in general, computationally prohibitive to resolve all of these scales in a simulation. Thus, an alternative approach is required to model the smaller scales of turbulent motion which are not captured by the discretisation method. Such an approach is referred to as turbulence modelling. A simulation resolving all the scales of motion is referred to as a direct numerical simulation (DNS) and is the most accurate form of simulation possible – DNS results are used as a reference to assess model performance. Depending on the range of scales resolved by the simulations, and equivalently, the range of scales modelled by a model, the classification of the turbulence model can be done. Large eddy simulations resolve most of the turbulent scales while modelling the smallest scales removed from the flow by a filtering mechanism. In the Reynolds-Averaged Navier-Stokes equations, which is the oldest approach to turbulence modelling, the time-averaged velocity field is obtained by simulating the time-averaged governing equations where the contribution of the fluctuation components are expressed through additional terms known as Reynolds stresses which need to be modelled. A more detailed study of turbulence modelling is presented in chapter 2 elaborating on LES and RANS models as well as other methods available in literature.

**Experimental fluid dynamics**

**A brief history of EFD**

A succinct and well-written history of the major contributors and their contributions to fluid dynamics, as we know it, is provided by Anderson (2010). Here, I summarise in brief the contribution related to the field of experimental developments for fluid flow measurement. The first diagram of flow streamlines around an object dates back 500 years to Leonardo Da Vinci. He also stated vaguely the continuity equation for low-speed flows as well as stating the ’wind-tunnel principle’ – the relative flow over a stationary body mounted in a wind-tunnel is the same as the relative flow over the same body moving through a stationary fluid. Following this, experiments on fluids were conducted on fluid statics during the 17-18th century. An important advancement to the field of EFD is the invention of the Pitot tube by Henri Pitot in the year 1732. The Pitot tube is capable of measuring the flow velocity at a point within the fluid and is still a commonplace equipment in any fluids laboratory. The study of turbulent flows is attributed to the famous experiments of Osborne Reynolds who studied the flow of fluids in a pipe and the eﬀect of varying parameters on this flow by releasing a dye in the stream. His study on the flow variation with parameters led to the famous non-dimensional Reynolds number used to segregate a flow into laminar or turbulent. Another prominent example in the field of flow measurement is the experiments of Prandtl in 1904, where he studied the boundary layer as well as flow separation characteristics behind solid bodies in a flow using a water channel. Since the experimental attempts of Reynolds and Prandtl, technical advances in the associated fields of optics, computers, electronics, image and signal processing, among others, has revolutionised measurement techniques for EFD. A few established techniques are discussed in successive sections.

**Hot wire anemometry (HWA)**

HWA was first introduced in the early 20th century before being commercialised mid-century and is one of the fundamental tools used for measuring point-wise velocity fields especially in turbulent flows. HWA calculates velocity based on the amount of cooling induced by the flow on a heated sensor maintained at a constant temperature by a feedback-control loop. The voltage drop across the sensor required to maintain equilibrium is used to measure the instantaneous velocity of the flow. As the velocity over the sensor changes, the corresponding convective heat transfer coeﬃcient varies leading to a diﬀerent voltage drop necessary to maintain equilibrium. Figure 1.3 depicts the electrical circuit diagram for HWA. The heat transfer from the sensor is not just a function of the flow velocity but also its geometry, orientation, the flow temperature, and concentration. Thus, it is possible to measure with HWA other fluid parameters such as concentration (Harion et al., 1996), temperature (Ndoye et al., 2010), as well as the individual components of the velocity field. Newer designs of probes are capable of measuring simultaneously all three velocity components. In addition, HWA’s capability for high data-gathering rate of the order of several hundred kHz as well as the very small size of the sensor which minimises disturbance of the measured flow are distinct advantages of HWA over other intrusive flow measurement techniques. However, a stringent need for calibration, as well as the possibility of contamination/breakage due to fragility of the sensor limit application of HWA to all flows.

#### Laser doppler anemometry (LDA)

The Doppler eﬀect refers to the change in frequency or wavelength of a given wave signal when the relative velocity between the source and the observer varies. This shift is exploited in LDA to measure the velocity of a flow by illuminating it with a laser beam which are reflected/scattered by particles seeded into the flow. The frequency diﬀerence between the original incident beam and the scattered beam, the Doppler shift, is linearly proportional to the particle velocity. The non-intrusive nature of LDA along with the high spatial and temporal resolution of the method and its ability to perform multi-component measurements are intrinsic advantages of LDA. However, its application is limited to ar-eas with optical access, and single-point measurements. Seeding particles and seeding concentration are important factors which aﬀect the performance of LDA. Depending on the optical arrangement of the Laser and the receiver, LDA can be classified as forward-scattering, side-scattering, or back-scattering LDA with the last being the most preferred option in modern set-ups. One such commercial set-up for an LDA experiment is shown in figure 1.4 – the fringe mode of LDA measurements is employed here by splitting the laser into two beams using a Bragg cell which intersect in the measurement zone. The frequency shift between the two intersection laser beams, induced due to their diﬀerent incident angles, is used to calculate the Doppler frequency which is directly proportional to velocity.

**Table of contents :**

**1 Fluid flow fundamentals **

1.1 Computational fluid dynamics

1.1.1 A brief history of CFD

1.1.2 Governing equations

1.1.3 Discretisation techniques

1.1.4 Boundary conditions

1.1.5 Turbulence modelling

1.2 Experimental fluid dynamics

1.2.1 A brief history of EFD

1.2.2 Hot wire anemometry (HWA)

1.2.3 Laser doppler anemometry (LDA)

1.2.4 Particle image velocimetry (PIV)

1.3 Limitations

1.4 Coupling EFD and CFD

**2 Turbulencemodelling **

2.1 Reynolds averaged Navier-Stokes

2.1.1 RANS equations

2.1.2 Turbulent-viscosity models

2.1.3 Reynolds stress models (RSM)

2.2 Large eddy simulation

2.2.1 Filtered Navier-Stokes equations

2.2.2 Smagorinsky models

2.2.3 WALE model

2.2.4 Models under location uncertainty

2.2.5 Other Approaches

2.3 Hybrid approaches

**3 Large eddy simulation- applications **

3.1 Flow solver – Incompact3d

3.2 Channel flow at Re⌧ = 395

3.2.1 Channel flow parameters

3.2.2 Channel flow results

3.3 Wake flow around a circular cylinder at Re = 3900

3.3.1 Introduction to wake flow

3.3.2 Flow configuration and numerical methods

3.3.3 Wake flow results

3.4 Concluding remarks

**4 StochasticSimulations – vortex forces andLangmuir circulation **

4.1 Introduction

4.2 Mathematical formulations

4.2.1 CL model

4.2.2 Large-scale drift MULU

4.3 Vortex force theory for streak formation

4.4 Large-scale application

**5 Data assimilationtechniques **

5.1 Mathematical Formulation

5.2 Data assimilation – a review

5.3 4D-Var approach

5.3.1 4D-Var Algorithm Variants

**6 Flow reconstruction – snapshot optimisation method **

6.1 Introduction

6.2 Snapshot optimisation method

6.2.1 SO model concept

6.2.2 Mathematical formulation

6.2.3 Reduced order formulation

6.2.4 Averaging SO method

6.3 Application and results

6.3.1 Synthetic case-studies

6.3.2 Algorithm Enhancement Techniques

6.3.3 Experimental case-studies

6.4 Conclusion

**7 Data assimilation-wakeflowatRe=3900**

7.1 Code Formulation

7.1.1 4D-Var

7.1.2 4D-Varles

7.2 3D synthetic assimilation

7.2.1 Flow configuration

7.2.2 Observations construction

7.2.3 Initial background construction

7.2.4 Results and Discussion

7.3 2D synthetic assimilation

7.3.1 Flow configuration

7.3.2 Observations construction

7.3.3 Initial background construction

7.3.4 Results and Discussion

7.4 Background covariance

7.5 Coefficient estimation

7.6 Combined case study

7.7 Conclusions