Objective:- In this paper, we discuss the importance of turbulence as a physical phenomenon and describe the main features of turbulent flow that are easily recognized & make it distinguishable from laminar flow. We follow this by introducing the 3 Navier Stokes Equations i.e. the Continuity Equation, Momentum Equation & Energy Equation. This is followed by introducing Reynolds Decomposition equation & using it to insert the N-S equations to derive the Reynolds Averaged Navier Stokes Equations. In addition, we extract the Reynolds stress term from the above-derived equation.

           Keywords: Importance of the study of turbulence; definition of turbulence; characterization of turbulent flows; Governing equations or Navier stokes equations.

Introduction To Fluid Flow Regimes & Reynold's Experiment:-                                                                                                                                                                                                            

 In fluid dynamics, a flow can be Laminar, Turbulent, or Transitional in nature. This becomes a very important classification of flows and is brought out vividly by the experiment conducted by Osborne Reynolds. An experiment was conducted in which a flow through a glass tube Fig 1.1, he injected a dye to observe the nature of the flow. When the speeds were small the flow seemed to follow a straight-line path (with a slight blurring due to dye diffusion). As the flow speed was increased the dye fluctuates and one observes intermittent bursts. As the flow speed is further increased the dye is blurred and seems to fill the entire pipe. These are what we call Laminar, Transitional, and Turbulent Flows. Turbulence or turbulent flow is a type of fluid motion that is characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between those layers. A pictorial representation of a laminar flow transitioning into turbulent flow is shown in Fig 1.2. It is also found that flow in a pipe is laminar if the Reynolds Number (based on the diameter of the pipe) is less than 2100 and is turbulent if it is greater than 4000. Transitional Flow prevails between these two limits. At this juncture it may be mentioned that in a flow which is laminar, the flow variables like velocity components, pressure, temperature etc., have either constant values at a point in the flow field or the flow variables show a known variation with time. On the other hand, when the flow is turbulent, the flow variables display random variations with time and the fluctuations are three dimensional in nature. Further, turbulent flows have higher diffusion and dissipation. Hence, the transfer of heat, mass, and momentum is faster in turbulent flows. At the same time, the pressure loss and drag are also higher in these flows. The Reynolds number is defined as

where:

• ρ is the density of the fluid (kg/m³)
• v is a characteristic velocity of the fluid with respect to the object (m/s)
• L is a characteristic linear dimension (m)
• μ is the dynamic viscosity of the fluid (Pa·s or N·s/m² or kg/(m·s)).

                                                                                                                  Fig 1.1:- Reynolds Experiment. 

                                                                 Fig 1.2:- A representation of the transition from laminar (smooth) flow to turbulent flow. 

Turbulent flows occur in many situations of practical interest. For example flows (i) past vehicles like airplanes, cars, and ships (ii) in pipes, ducts, and process equipment (iii) in internal combustion engines, (iv) in the atmosphere, etc. are turbulent. Hence, the study of turbulence is of primary interest in aerospace engineering, chemical engineering, civil engineering, mechanical engineering, metallurgy, meteorology, ocean engineering, and other branches of engineering dealing with fluid flow. 

Turbulence is commonly observed in everyday phenomena such as surf, fast-flowing rivers, billowing storm clouds, or smoke from a chimney, and most fluid flows occurring in nature or created in engineering applications are turbulent. Turbulence is caused by excessive kinetic energy in parts of a fluid flow, which overcomes the damping effect of the fluid's viscosity. For this reason turbulence is commonly realized in low viscosity fluids. In general terms, in turbulent flow, unsteady vortices appear of many sizes that interact with each other, consequently drag due to friction effects increases. This increases the energy needed to pump fluid through a pipe. Turbulence can be exploited, for example, by devices such as aerodynamic spoilers on aircraft that "spoil" the laminar flow to increase drag and reduce lift.

Important Characteristics of Turbulent Flow:-                                                                                                                                                                                                                                       

A turbulent flow can be expected to exhibit all of the following features:

• Disorganized, chaotic, seemingly random behavior.
• Non-repeatability (i.e. sensitivity to initial conditions).
• Enhanced diffusion (mixing) and dissipation (both of which are mediated by viscosity at molecular scales).
• Three-dimensionality, time dependence, and rotationality.

Turbulence Modelling:-                                                                                                                                                                                                                                                                                

While computation it should be noted that (a) the turbulent flows are governed by Navier-Stokes equations, (b) turbulent flows can be considered as consisting of a large number of eddies of different sizes and the ratio of the length scale of the largest eddy to the smallest eddy is very large. Keeping in view all these features, it is evident that the computation of turbulent flow in all its details, called Direct Numerical Simulation (DNS), involves a solution of three dimensional, unsteady Navier-Stokes equations with a fine grid resolution. Since the grid is fine, the time step in computation would also be very small. Thus, DNS is computation-intensive from computer time and memory points of view. An alternate way is to decompose the flow parameter, r, say U, as (ū+u') where ū is the average of U over a time interval T and u' is the fluctuating part as shown in Fig 1.3. When (ū+u') is substituted for U in the Navier-Stokes equations and time average is taken, which results in a set of equations called Reynolds averaged Navier-Stokes (RANS) equations. These equations involve averages of products of fluctuating quantities. These are additional unknowns. To make the RANS equations a closed set of equations (i.e. number of equations equal to the number of unknowns), these additional terms need to be modeled. This aspect of the study of turbulence is called turbulence modeling. For this particular paper, we will be making use of the Reynolds decomposition method & insert the (ū+u') equation in the N-S equation to derive the RANS equation. 

                                                                                                             Fig 1.3:- Typical turbulent signal &  time averaging. 

Navier Stokes Equations:-                                                                                                                                                                                                                                                                          

The three-dimensional unsteady form of the Navier-Stokes Equations is expressed below. These equations describe how the velocity, pressure, temperature, and density of moving fluid are related. The Navier-Stokes equations consist of a time-dependent continuity equation for conservation of mass, three time-dependent conservation of momentum equations, and time-dependent conservation of energy equation. There are four independent variables in the problem, the x, y, and z spatial coordinates of the same domain, and the time t. There are six dependent variables; the pressure p, density ρ, and temperature T and three components of the velocity vector i.e. u component is in the x-direction, the v component is in the y-direction, and the w component is in the z-direction. 

                                                        Continuity Equation:- `(del rho)/ (del t) + (∂(rho u))/ (∂ x) + (∂(rho v))/( ∂ y) + (∂(rho w))/( ∂ z) = 0`

            Momentum equation along the x-direction:- `(del (rho * u))/(del t) + (del(rho * u^2))/(delx) + (del (rho * u * v))/(del y) + (del (rho * u * w))/(del z) = - (del p)/ (delx) + 1/(Re) * { (del tau x x)/(del x) + (del tau x y)/(del y) + (del tau x z)/(del z) }`

           Momentum equation along the y-direction:- `(del (rho * v))/(del t) + (del(rho * v^2))/(dely) + (del( rho * u * v))/(del x) + (del (rho * v * w))/(del z) = - (del p)/ (dely) + 1/(Re) * { (del tau x y)/(del x) + (del tau y y)/(del y) + (del tau y z)/(del z) }`

          Momentum equation along z-direction:- `(del (rho * w))/(del t) + (del(rho * w^2))/(delz) + (del( rho * u * w))/(del x) + (del (rho * v * w))/(del y) = - (del p)/ (delz) + 1/(Re) * { (del tau x x)/(del x) + (del tau y z)/(del y) + (del tau z z)/(del z) }`

The terms on the left-hand side of the momentum equations are called the convection terms of the equations. Convection is a physical process that occurs in a flow of gas in which some property is transported by the ordered motion of the flow. The terms on the right-hand side of the momentum equations that are multiplied by the inverse Reynolds number are called the diffusion terms. Diffusion is a physical process that occurs in a flow of gas in which some property is transported by the random motion of the molecules of the gas. Diffusion is related to the stress tensor and to the viscosity of the gas. Turbulence and the generation of boundary layers are the results of diffusion in the flow. 

Derivation of RANS equation:-                                                                                                                                                                                                                                                                   

The basic tool required for the derivation of the RANS equations from the instantaneous Navier–Stokes equations is the Reynolds decomposition. Reynolds decomposition refers to the separation of the flow variable (like velocity u) into the mean (time-averaged) component () and the fluctuating component (). Thus,

                                                                                               where  is the position vector.

The following rules will be useful while deriving the RANS. If f and g are two flow variables (like density (ρ), velocity (u), pressure (p), etc.) and s is one of the independent variables (x,y,z, or t) then,

 Now the Navier–Stokes equations of motion for an incompressible Newtonian fluid are:

Substituting,

                                                                                         , etc

and taking a time-average of these equations yields,

The momentum equation can also be written as, 

On further manipulations this yields,

where,  is the mean rate of the strain tensor.

Finally, since integration in time removes the time dependence of the resultant terms, the time derivative must be eliminated, leaving:

The true-time average () of a variable (x) is defined by

For this to be a well-defined term, the limit () must be independent of the initial condition at t₀. In the case of a chaotic dynamical system, which the equations under turbulent conditions are thought to be, this means that the system can have only one strange attractor, a result that has yet to be proved for the Navier-Stokes equations. However, assuming the limit exists (which it does for any bounded system, which fluid velocities certainly are), there exists some T such that integration from t₀ to T is arbitrarily close to the average. This means that given transient data over a sufficiently large time, the average can be numerically computed within some small error. However, there is no analytical way to obtain an upper bound on T.

By definition, the mean of the fluctuating quantity is zero                               ().

Some authors prefer using U instead of  for the mean term (since an overbar is used to represent a vector). Also it is common practice to represent the fluctuating term  by u, even though u refers to the instantaneous value. This is possible because the two terms do not appear simultaneously in the same equation. To avoid confusion we will use  to represent the instantaneous, mean, and fluctuating term.

The equations are expressed in tensor notation, which greatly simplifies the maths.

Time-averaging these equations yields,

Note that the nonlinear terms (like ) can be simplified to,

This follows from the mass conservation equation which gives,

Reynolds - Averaged Navier Stokes Equation (RANS):-                                                                                                                                                                                                                               

The Reynolds-averaged Navier–Stokes (RANS) equations are time-averaged equations of motion for fluid flow. They are primarily used while dealing with turbulent flows as mentioned earlier. These equations can be used with approximations based on knowledge of the properties of flow turbulence to give approximate averaged solutions to the Navier–Stokes equations. For a stationary, incompressible flow of Newtonian fluid, these equations can be written in Einstein notation as:

The left-hand side of this equation represents the change in mean momentum of fluid element owing to the unsteadiness in the mean flow and the convection by the mean flow. This change is balanced by the mean body force, the isotropic stress owing to the mean pressure field, the viscous stresses, and apparent stress  owing to the fluctuating velocity field, generally referred to as the Reynolds stress. This nonlinear Reynolds stress term requires additional modeling to close the RANS equation for solving and has led to the creation of many different turbulence models.