which description would you choose describe turbulence best:

Big

whirls have little whirls that feed on their velocity,and little whirls have lesser whirls and so on to viscosity…Lewis Fry Richardson

- Turbulence is a phenomenon which sets in in a viscous fluid for values of viscosity, hence its purest, limiting form may be interpreted as asymptotic, limiting behavior of viscous fluid as its coefficient approaches zero… Neumann
- Turbulence is a manifestation of the spatial and temporal chaotic behaviour of fluid flows at large Reynolds numbers, of a strong ly nonlinear dissipative system with an enormous large number of degrees of freedom described by th Navier-Stokes equations.

Or perhaps none is sufficient to say one of the above as conclusive description of the **quantitative** features manifested in the turbulence phenomenon.

The first bold quest to describe turbulence by means of convenient understanding where those conducted in the seminal work of Tenneks and Lumley, and to do so in a **qualitative** manner.

A First Course in Turbulence (MIT Press)

So let us begin:

Turbulence is without a doubt chaotic. One of the most important features the is**Intristic spatial-temporal seemingly random irregularity:***self -stochatization.*Meaning there is no need for an external random forcing in the fluid domain its self or at its boundaries and no initial condition is of necessity as far as the reynolds number – a dimensionless number reflecting on how well momentum is diffused relative to the flow velocity (in the cross-stream direction) and on the thickness of a boundary layer relative to the body which is “large enough”.

An interesting question raised by J. K. McDonough is how such behaviour arises from a set of deterministic equations as NSE. The answer lies in the equation being extremely sensitive in initial, boundary conditions or external noise. Turbulence acts as the ultimate non-linear oscillator and an amplifier with an enormous gain.In atmospheric flows, relevent scales range from hundreds of kilometers to sizes comparable with less than a millimeter. Putting this on somewhat of a mathematical ground this translates to about 10^30 excited degrees of freedom is which are strongly interacting. This fact is extremely interesting, again contributing a J. M. Mcdonough standpoint that the**Wide range of strongly interacting scales:***statistical description*is by no means statistical*theorization*(consider this strong point when conducting CFD by statistical measures such as RANS and LES).

The strong interaction between som many scales resulting from non-linearity (linear systems with as many scales possible do not excite each other).*Wide range of interacting scales*Chaos is chaos. Two initially nearly (but not precisely, let us say to a difference sizable of the order of 10^-30 mm)become unrecognizable different on a time scale of engineering or even academic interest. One realization is strongly different from the other (consider this fact as quantitive trying to validate a DNS by one realization. Strongly chaotic systems like turbulence are extremely sensitive to small disturbances. The interesting concept which qualitatively wins the day is that different realizations of the same turbulent flows**Unpredictability:***have the same statistical properties.*

That is almost al statistical properties carry that special and beloved feature, which actually means that statistical features of different realizations of the same turbulent flows are insensitive to disturbances –*they are statistically stable.*Mind you not, this feature isThis means that turbulent flows carry both predictable and unpredictable features. different pairs of realizations have the same statistical features.**only statistical (CFD statistical methods rely on this for “simulating reality”).****The qualitative route wins again.**It is a well-known fact that a source of energy is an inseparable ingredient to maintain turbulence, and the energy is supplied mostly by the large scales while the dissipation occurs at the small ones. statistically irreversible process is happening – turbulent flows are directed only in one way in time.**Turbulent flows are highly dissipative:**The “random” fields of vorticity (curl of the velocity vector) own predominant vortex stretching. This actually means continuous positive production of enstrophy by nonlinear inertial processes, that dissipate by viscosity. A feature exhibited is positive production of strain due to amplification of the gradients of the velocity fields (as both strain and rotation are an extraction from the velocity gradient tensor). The direct three-dimensional rotational nature is a qualitative feature extracted from realization of DNS (especially by visualisation of the Q factor).**Turbulent flows are three-dimensional and rotational**

Turbulent Boundary Layer DNS (APS Gallery Submission)

Strongly and enhanced transport processes are exhibited in turbulent flows. Besides momentum and energy these are the passive objects (scalars like particles, heat and moisture and vectors liken gradients of passive scalars and magnetic fields).**Strongly diffusive :**

The enhancement of diffusive effects is not actually a particular property of turbulence as every random field, even Lagrangian chaotic laminar flows exhibit enhanced transport of passive objects.

Particle image velocimetry (PIV) combined or better yet 3DPTV (stand corrected by one of my mentors in a comment below) with an Eulerian/Lagrangian DNS are the future arsenal of weaponry to explore quantitative manifestation of the well exhibited qualitative feature.

*particle image velocimetry*

To Conclude, these widely known qualitative features of all turbulence flows are the same in essence. By that we may keep talking about the concept of qualitative universality.

We shall close with one of the videos I found as a pearl:

]]>Momentum Equation in fluid dynamics is a way of expressing Newton’s Second Law of Motion in differential form for a control volume. The same goes with the Energy equation which simply expresses the First law of Thermodynamics in differential form for a control volume.

These equations of mechanics and thermodynamics such as Newton’s second law are originally described for Particle or Control Mass systems, not for the control volume approach. It is described for **Lagrangian **description. For example, if we identify the trajectory of each particle then we can write the trajectory of those particles in mathematical form using Newton’s Second Law of motion.

On the other hand, We can sit and focus a confined space, Particles enter and leaves that confined space. We don’t have to track the particle but we are interested to know what’s happening to those particles in that confined space. This approach is known as ** Eulerian approach** or

Since the fluid is continuously deforming, it is more convenient for us to study fluid dynamics using the Eulerian approach rather than tracking each fluid particle trajectory using the Lagrangian approach.

*The problem is that the basic equations of mechanics and thermodynamics are not originally developed for Eularian Description.*

So we must do the transformation from Lagrangian to Eulerian description so we can use these basic laws of mechanics and thermodynamics that were originally developed in the Lagrangian frame. This transformation is given by well known * Reynolds Transport Theorem*.

The above equation is nothing but ** Reynolds Transport Equation**.

RRT expression is a mathematical representation of the ** Conservation principle**. RRT can be used to derive

To derive the Linear-Momentum Conservation equation, We will consider * N* to be

Substituting that in our general Reynolds Transport Equation and assuming non-deformable and stationary control volume dv, We can obtain the equation shown below.

Let’s focus on the LHS of the above equation. We can apply Newton’s second law of motion for a system of particles but not for a control volume directly. If we apply Newton’s second law of motion then LHS is equal to the Total resultant force on the system. Derivation of Reynolds transport equation involves delta(t) tending to 0 (t – time) situation such that system and control volume almost merges/coincides into each other. Hence we can say Resultant force acting on the system is equal to Resultant force acting on the control volume.

Reynolds Transport theorem is used for inter-conversion between system and control volume. Using RTT we can effectively write Newton’s Second Law of motion for control volume which is nothing but the principle of Linear Momentum.

** Above equation is nothing but Conservation of Linear Momentum in Integral form**.

We can not solve the above Linear Momentum equation because we have not specified any description on Forces so the equation is not closed. In continuum mechanics, we can classify force as Surface and Body force.

**Body Force** – Body force is fairly easy to understand and implement mathematically. Body force acts over the volume of fluid elements. It does not require any contact for its action rather it requires an external field. For example Gravity Force, Magnetic force etc.

**Surface Force** – Surface force on a fluid is a little tricky to implement mathematically as compared with Body Force. Surface force requires contact on control volume through control surface for its action. Foe example Pressure force, Shear force etc.

To understand Surface force in detail, Lets introduce the concept of ** Traction Vector**.

**Surface forces** can be represented in terms of **force per unit area** and formally** Traction Vector **is used to represent surface force on any **arbitrarily oriented area** in a fluid element. The surface force will not only depend on the choice of location of the area but it will also depend on the choice of the orientation of the area.

**Our goal** is to relate this Traction vector which is applied on the arbitrarily oriented surface to the surface which has standard orientation i.e. having direction normal along x, y and z-axis of the Cartesian coordinate system because the Cartesian coordinate system is more often used to solve general fluid flow problems. These surfaces are special surfaces on which we have the special effect of the traction vector. We will try to understand that by looking at the simple diagram shown below**.**

For these surfaces, we have an equivalent notation of Traction Vector using **Tau**.

Ta**u _{i,j}** is formally called a Component of

This Stress tensor vector can have 9 components in the Cartesian coordinate system as the i index varies from 1 to 3 and the j index also varies from 1 to 3.

*But we have 6 independent components of the Stress tensor vector not 9 by the conservation of angular momentum.*

Now those 6 independent components of Stress Tensor Vector can only be used for the surfaces that are normal to x, y and z-axis. What happens when the surface is arbitrary oriented. For most practical applications, geometries inside which fluid is flowing are complex and have arbitrarily oriented surfaces. We must relate the Stress tensor vector to the Traction Vector as the Traction vector is used to specify surface force on arbitrarily oriented surfaces. Remember that was our initial goal to find out what happens to the Traction vector when it encounters the surface which has orientation along the x, y & z-axis of the Cartesian coordinate system. That’s what we are going to discuss in the next section.

To relate the Stress Tensor vector to the Traction vector we will use a special type of fluid volume as shown below :-

Fluid volume is shown above is very special in one sense because it has 3 surfaces PCB, PCA & PAB are normal to x_{1} (x), x_{2} (y) and x_{3} (z) directions respectively and the fourth surface ABC has an arbitrary orientation.

Surface forces on the PCB, PCA & PAB can be specified by Stress Tensor Vector and surface force on the surface ABC can be specified by using Traction Vector. We can specify surface force along x, y and z-direction for all surfaces. Then we can write the * equation of equilibrium* for this volume then the surface force for which orientation of normal is arbitrary can be expressed in terms of forces on the surface for which has normal direction along x, y and z-direction. In this way, we can relate arbitrary Traction vectors in terms of Stress Tensor Vector components.

We can note that dA_{1} is a projection of dA_{n} on x_{2}-x_{3} plane. Similarly, dA_{2} is a projection of dA_{n} on x_{1}-x_{3} plane and dA_{3} is a projection of dA_{n} on x_{1}-x_{2} plane. Hence we can write:-

After equation above expression in our equation of equilibrium we can obtain the equation shown below :-

*Above equation shows the Traction vector as a fuction of stress tensor vector.*

*j is the repeated index which is also called a dummy index and when we have a repeated index it means we have an invisible summition. *

**Above expression is known as Cauchy Theorem.**

relatesCauchy TheoremTraction vectoon a surface having arbitrary normal with theron a surface that has normal to a reference known (Cartesian) plane.Stress tensor vector

*In Matrix Notation we can write:-*

**Stress Tensor Vector components are mapping the Normal vector on to the Traction Vector.**

This is one of the special property of Second order Tensor that it maps a vector on to an another vector.

Going back to our Equation of Linear Momemtum conservation.

*For x – direction, Linear Momentum equation will be :-*

Let’s consider an Arbitrary Control volume **V/dV** having arbitrary aligned surface **dA/dS**. We can calculate the total force on that arbitrary control volume based on the above understandings and discussions.

**We can apply the Divergence theorem on the surface force term to convert surface integral into volume integral.**

**Doing so, we can obtain as shown below**

We can substitute Total Force on CV expression into the equation of Linear Momentum conservation. Doing that we can obtain as shown below.

We can covert the second term on LHS into volume integral by using the ** Divergence t**heorem.

Doing that, then taking all the RHS terms on LHS & taking volume integral common we can obtain as shown below :-

Choice of control volume **dv** is arbitary hence to have volume integrand equal to 0, function inside integral should also be zero. Keeping that in mind, we can finally obtain :-

** Last 2 equation shown above are same but just written in index notation!**

Abov equation is nothing but ** Navier-Cauchy Equation or Navier-Equation of Equilibrium**.

This is the well-known * Navier-Cauchy Equation or Navier-Equation of Equilibrium*. In a simple word, it is a mathematical representation of Newton’s Second Law of Motion written for a fluid material in a control volume space. On LHS we have Unsteady & Convection term which if simplified (with the help of continuity equation) can be shown as a product of fluid density & total derivative of fluid acceleration which is nothing but the rate of change of fluid momentum that’s equal to RHS where we have Surface and Body Force which is what Newton’s Second Law of Motion talks about!

Complete mathematical description of surface force term is still pending in Navier-Cauchy Equation. For specific type of fluid, the surface force can be mathematically described which will eventually lead to the famous ** Navier-Stokes Equation.** Let’s keep that discussion for the

Transport equations for each velocity component – momentum equations – can be derived from the general transport equation by replacing the variable φ with u, v and w respectively.

The above equations govern a two-dimensional laminar steady ﬂow.

*The velocity ﬁeld obtained from the momentum equation must also satisfy the continuity equation.* *The convective terms of the momentum equations contain **non-linear** quantities: for example, the ﬁrst term of the equation is the x derivative of ρu ^{2}.*

*All three equations are intricately coupled because every velocity component appears in each momentum equation and in the continuity equation.*

*The most complex issue to resolve is the role played by the pressure. It appears in both momentum equations, but there is evidently no (transport or other) equation for the pressure.*

If the pressure gradient is known, the process of obtaining discretized* equations for velocities from the momentum equations is exactly the same as that for any other scalar.*

*If the ﬂow is **compressible** the continuity equation may be used as the transport equation for density and, the energy equation is the transport equation for temperature.* *The **pressure** may then be obtained from density and temperature by using the equation of state p = p(ρ, T).*

*What happens when the flow is incompressible?*

*If the ﬂow is **incompressible** the density is constant and hence by deﬁnition not linked to the pressure.* *In this case **coupling between pressure and velocity** introduces a constraint in the solution of the ﬂow ﬁeld.* *We don’t have any separate equation for pressure.*

Still,we need to supply the correct pressure field into the momentum equation so that theresulting velocity field satisfies the continuity constrain.

*The pressure–velocity linkage can be resolved by adopting an iterative solution strategy such as the **SIMPLE algorithm of Patankar and Spalding (1972).*

*In SIMPLE Algorithm:-*

ØThe convective ﬂuxes through cell faces are evaluated from guessed velocity components.

Ø A guessed pressure ﬁeld is used to solve the momentum equations.

ØA **pressure correction equation **is derived from the continuity & momentum equation.

ØThat Pressure Correction equation is solved to obtain a pressure correction ﬁeld, which is in turn used to update the velocity and pressure ﬁelds that will satisfy the continuity equation.

ØAgain Momentum equation is solved with the updated pressure and velocity field.

ØThe process is iterated until convergence of the velocity and pressure ﬁelds.

*Based on the premise that fluid flows from regions with high pressure to low pressure*.

*The SIMPLE stands for **Semi-Implicit** **M**ethod for **P**ressure- **L**inked **E**quations.*

The algorithm is essentially a ** guess-and-correct procedure **for the calculation of pressure on the staggered grid arrangement.

Now, the steps involved in solving the incompressible Navier Stokes Equation using SIMPLE Algorithm are explained.

*2. Guess & Solve*

Guess a pressure ﬁeld **p* **is guessed.

Discretised momentum equations are solved using the guessed pressure ﬁeld to yield velocity components u* and v* as follows:

**u* **& **v* **satisfies the momentum equation but does not satisfy the continuity equation.

The equation shown above is obtained from Finite Volume Discretization of Momentum Equation.

*2. Introduce a Correction Term*

We deﬁne the correction **p′ **as the difference between correct **pressure ﬁeld p **and the **guessed pressure ﬁeld p***.

Similarly, we deﬁne **velocity corrections u′ and v′ **to relate the **correct velocities u & v **to the **guessed velocities u* and v*.**

*3. Obtain Velocity Correction*

*3. Obtain Velocity Correction*

*4. Obtain an Equation for Pressure Correction P’*

As mentioned above, Momentum Equation solved using guessed pressure field satisfies the momentum conservation but not the mass conservation.

Then we introduce a correction term **P’, U’, V’ **which is added with guessed value **P*, U*, V* **to correct the pressure and velocity field.

Then we obtain an expression that corrects the velocity field based on the guessed velocities **U’, V’** and corrected pressure **P’ **as shown below.

Now we have to obtain an equation for pressure correction term **P’ **so that we can correct our velocity field.

*5. Correct the Pressure Field*

*6. Correct the Velocity Field*

Now, we have to find some way to put information back into momentum equation for example strong mixing process that turbulence motion has on the mean flow.

That’s where Turbulence Modelling comes into the picture.

It is very important to highlight that, It is a very strong simplification because all of the turbulence information is lost by averaging.

These equation are known as Reynolds Average Navier Stokes Equation. We have an additional term in the right most corner. It’s unit is same as the unit of stress hence it is know as ** Reynolds Stress**. We can see the expanded form of Reynolds Stress term.

We can write Reynolds Stress term in a Matrix Form.

** Diagonal Terms **are known as

** Off Diagonal Terms **are known as

Originally we have 9 unknowns – 3 diagonal and 6 off diagonal element. But, The Reynolds stress tensor matric is ** symmetric**. Meaning, Three terms appear above the diagonal elements are same as Three terms appear below the diagonal element. Hence, The total number of unknowns are actually six i.e. 3 along the diagonal and 3 off diagonal.

This is an another way of writing RANS equation as shown above. On LHS we have **total acceleration **written in averaged quantity. On RHS we have **pressure gradient term, laminar contribution of stress **and **turbulence contribution of stress**. Main task in Turbulence Modelling is to model these additional stress terms. Turbulence flow exhibits random fluctuation both in time and space. So it is difficult to predict those fluctuations. But, Statistical way of dealing is possible.

Replacing the Reynolds stresses with an appropriate quantity which can be modelled mathematically is known as Turbulence Modelling.

** II term Mean Viscous Stress Tensor **– is due to laminar viscosity/dynamic viscosity. It is mean viscous stress tensor because mean velocity quantities are used.

** III term Reynolds Stress Tensor **– is due to averaging procedure that we apply on the

We have 3 Momentum and 1 Pressure Equation. When the flow is laminar we have 4 variables x, y , z velocity and pressure. Hence Problem is closed. In the Turbulence flow problem, We have additional unknown stress term in our momentum equation. But, the number of equations are still 4. Hence no. of variables are not equal to number of variables hence *Problem remains unclosed.*

** Now to close the problem either one has to get additional equation or replace the unknown variables by suitable known variable.** We can derive the additional equations to model our Reynolds Stresses but it will end up in adding more unknowns. The alternative way is to replace the unknown terms with known variables. This process is known as

In 1887, Boussinesq proposed that the ** Reynolds Stresses (τ_{ij}) **can be related to the

Based on the dimensional argument, Eddy Viscosity is proportional to the *product of density, eddy velocity scale (*V_{t}*) and eddy length scale (*l_{t}*) as shown above.**Between Eddy length, time and velocity scale, If any two quantity are known then third quantity could be computed.* *Here eddy viscosity *μ_{t}* is a flow property not a fluid property.*

Momentum exchange through turbulence eddies. Largest eddies responsible for mixing. They have a time scale (T – turn-over time) and a length scale L. Now we need **two additional equations to describe L and T **so that we can compute eddy viscosity. Once the eddy viscosity is computed, we can put it back into boussinesq formula and calculate the unknown Reynolds Stress. In this way we can close the RANS equation.

The most natural framework to compute the two independent scales required for the eddy-viscosity **μ _{t} ~ L^{2}∕T ~ k^{2}∕ε ~ k∕ω**. One turbulence scale is computed from the

From ** turbulent kinetic energy **we can get

At first, We will start with obtaining equation of Reynolds Stresses.

**To do that, We can take difference of Instantaneous Navier-Stokes equation and Reynolds Averaged Navier-Stokes Equation.**

By taking the difference, We can obtain **Equation of **** Reynolds Stresses **as shown below:-

In order to reduce the unknowns, We will follow one step of tensor algebra that is known as *contraction.***Contraction** is nothing but taking the *trace or the isotropic part of the Reynolds Stress Tensor Matrix.**Isotropic Part is nothing but Diagonal Element of the Reynolds Stress Tensor Matrix.**If we make i = j, We can obtain Isotropic Part.*

Turbulence Kinetic Energy (k) is the ** mean kinetic energy per unit mass associated with eddies in turbulent flow.** Physically, the turbulence kinetic energy (k) is characterized by root-mean-square (RMS) of velocity fluctuations. Turbulence Kinetic Energy (k) only contains the

*After adding **all the equations of normal stresses we will get an **Equation for Turbulent Kinetic Energy, *k* as shown below.*

**Unsteady Term: **How the TKE at a certain point changes with time.

**Convection Term: **How the TKE is convected from one point to another by the mean flow velocity.

**Production Term: **Most important one, It has an interaction between turbulent fluctuation which is ** Reynolds Stress times the mean velocity gradient**, So the turbulence extracts the energy from the mean flow.

**Dissipation Term:** Here we have molecular viscosity times velocity gradient square. This term takes out the energy. It dissipates energy into heat.

**Turbulent Diffusion Term: **It is one of most complex term because it has triple product of fluctuating velocities and product of velocity and pressure. But the advantage here is this term is under the divergence, What that means is, if we integrate this term across the mixing layer, The integration of this term outside the mixing layer is zero as turbulence variable are zero outside the shear layer. So this term does not produce or destroy anything. It just distributes the energy slightly inside the layer. *For example: If we have a mixing layer or jet flow, This term will take some energy from some place and diffuse it to another place that’s why it is called as turbulent diffusion term.*

**Molecular Diffusion Term:** This term is the classical gradient diffusion or laminar diffusion term which is viscosity time gradient of turbulent kinetic energy.

This lead to famous 2 Equations Turbulence Model, K-Epsilon or K-Omega Turbulence Model.

Reynolds averaging is one of the approaches used to eliminate the turbulence scales. The application of this approach leads to the Reynolds Averaged Navier-Stokes (RANS) equations.

The Reynolds stress terms in the RANS equation require modelling in order to obtain a closed system of equations. The Boussinesq hypothesis is one of the key elements of turbulence modelling. The eddy-viscosity can be computed if two independent scales are available.

This leads naturally to two-equation turbulence models. Turbulence Kinetic Energy (k) is the mean kinetic energy per unit mass associated with eddies in turbulent flow.

Turbulence Kinetic Energy (k) only contains the normal/isotropic part of the Reynolds Stress Tensor.

The ** shear component or anisotropic part** of the

Most of the cfd applications consist of stationary objects around/inside which fluid will be flowing. So meshes are stationary. If the flow is incompressible, ** Incompressible Steady Navier-Stokes** equations are solved. Very common applications are – Flow around Aerofoil, Flow inside Pipe. In both cases aerofoil and pipe are stationary hence meshes are stationary & solved using Incompressible Steady Navier-Stokes equations.

But there are certain applications involving motion of the geometry for example turbo machines. In that case mesh motion is required. It could be rotation or translation. Now our problem is unsteady because of the motion of the mesh. At each and every time step mesh is being moved. Now we will be solving *Incompressible*** Unsteady Navier-Stokes **equations which are more computationally expensive to solve ofcourse.

Problem comes when we have to deal with **very high speed rotation or translation**. We need to keep “time step size” very small such that any flow variable does not jump more than one/two mesh cell in one time step.

For example if there is a turbine rotating at 36000 rpm and one would want to solve for 10 revolutions. So in this case 600 revolutions are happening in 1 seconds. 1 revolution will happen in 0.0016 second. For 10 revolutions, We have to solve totally for 0.0167 seconds. This is our **total solution time**, 0.0167 second.

Here we have focus on this – **1 revolution is happening in 0.0016 seconds. **In order to capture full transient flow physics we have to keep time step extremely small. For example let say it is only allowed to mesh move 10 degree per time step based on the accuracy requirements. So in this case time step will be 0.000044 seconds which is extremely small. So we can see **how computational expensive** situations can become.

What about **initial condition. **If the given initial conditions are poor. We have to further increase total solution time so that error in initial solution can be eliminated. Some time poor initial condition can lead to divergence also.

Other issue is when there is mesh in motion. Interface between moving and stationary region will generally going to be **non-conformal. **In order to transfer solution from moving region to the stationary region, CFD solver have to do some extra interpolation which makes computation more **expensive and unstable**.

What about **initial condition. **If the given initial conditions are poor. We have to further increase total solution time so that error in initial solution can be eliminated. Some time poor initial condition can lead to divergence also.

Other issue is when there is mesh in motion. Interface between moving and stationary region will generally going to be **non-conformal. **In order to transfer solution from moving region to the stationary region, CFD solver have to do some extra interpolation which makes computation more **expensive and unstable**.

This is the concept of MRF, Governing equations are solved in a reference frame that is rotating or translating with the same speed of the rotating/translating geometry. Physically it means we are sitting on the moving body and seeing the flow field around it. This makes the flow field steady relative to the geometry.

If there is a turbine rotating and we are standing on the floor then flow field around turbine would be transient from our perspective.

Instead of watching rotating turbine from a distance, If we sit on turbine blade and rotate with the blade then the flow field around us or turbine would be steady from our perspective.

Another well known example is River & Boat. Boat is moving with fixed velocity. There is a bridge above river. If you stand on bridge and observe flow of river water, First water below bridge is stand still then boat comes and create disturbance. Then boat goes and disturbance of water goes down. Observer standing on the bridge, flow appears to him will be unsteady because at a given location below the bridge, river water speed in changing.

If a person sitting in boat, moving with the boat itself. Now with respect to that person, River water will not change and He will see river water as steady flow field.

Steady state problems in the moving frame are easier to solve then transient problem with moving mesh.

*This approach significantly reduce computational cost.*

*Now, Equation of Fluid Dynamics are defined with respect to Moving Reference Frame (MRF). We must account for additional accelerations terms which models the affect of fluid motion in the moving frame.*

*What if vector itself is moving in the Rotating Frame?*

*Then the total dA/dT is the dA/dT as we see from the stationary frame + due to the change in the direction of vector A which is Omega X vector(A).*

For Momentum Conservation (Navier-Stokes Equation), Our interest is Acceleration, Which is double derivative of displacement.

*Coriolis Acceleration – If we are translating in rotating frame, we will experience a lateral force.*

*Centrifugal Acceleration – Experience when sitting on rotating frame. *

*Translation Acceleration – If we are sitting in translating reference frame.*

*Rotational Acceleration/Euler Force – This is due to angular acceleration of reference frame.*

Strength of source term centrifugal acceleration will increase as we go away from the origin of MRF

Let’s look at the left-hand side of the momentum equation of Eqn [2], by taking into account Eqn [1] for the acceleration term:

First term on RHS can be expanded and written as :-

*Finally :-*

Incompressible Navier-Stokes equations in the rotating frame, in terms of relative velocities

We want to solve single set of navier stokes equation in entire domain but with extra source term in rotating region only. As we can see image above we define separate region and assign it as MRF. By applying vector identities and doing rearrangements we can arrive to the Navier-Stokes equations in the relative frame with absolute velocity.

*This “source term” is only applied to the region of MRF only.*

*In the convection term we have absolute as well as relative velocity.*

*In CFD codes of Finite volume method. We discretize the Convection term in usual way which is integrating the terms and applying Gauss’s divergence theorem & we arrive to face volume fluxes which is shown below.*

*Now we substitute this expression shown below into the face volume fluxes.*

*Finally we arrive at :-*

*First term on RHS is same as we would have as in general NV-Stokes Equation.*

*Second term “Flux Correction”, Physically when we are jumping from absolute frame to rotating frame we have to correct the fluxes in order to account for relative velocity as we would see if we were in rotating with the MRF.*

**References: **ANSYS Officials, openFOAM Foundation