There are certain fundamental principles of conservation that govern the implementation of CFD. These principles are Conservation of mass, momentum & energy. We will see how we can represent these principles evolve in a mathematical form that expresses the dynamics of fluid flow by preserving mass, momentum and energy conservation.
The Need of Reynolds Transport Theorem (RTT)
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 Control volume approach.
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. N is an extensive property (Mass, Momentum, or Total Energy) & n is any intensive property (extensive property per unit mass). Using the Reynolds transport theorem we are able to establish a relationship between Control System and Control volume approach. LHS of RTT is the physically total rate of change of any extensive property with respect to a system that is equal to RHS. In RHS we have 2 terms, the First term is the physical time rate of change of extensive quantity in fixed CV and the second term is Outflow – Inflow through control volume surface in simple words. We can see we are able to relate the System approach (based on the Lagrangian frame) to the Control volume approach (based on the Eulerian frame) with an additional term that represents net flow across control volume surface.
RRT expression is a mathematical representation of the Conservation principle. RRT can be used to derive Continuity Equation, Momentum Equation, Energy Equation, Species Equation etc.
The Linear-Momentum Conservation Equation Origin from RTT
To derive the Linear-Momentum Conservation equation, We will consider N to be Linear momentum of the system and n to be the Linear Momentum per unit mass of the system.
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.
The Forcing & Stress Tensor Concept in Fluid: Fluid Kinematics Perspective
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.
Taui,j is formally called a Component of Stress TensorVector. It is a Second-order tensor because it requires 2 indices i & j for its specification. Just like a Vector which is the first-order Tensor because it requires only 1 index for its specification. Scalar is a zeroth-order tensor because it does not require any index for its specification. Stress Tensor Vector is used for the specification of surface force on the fluid element.
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.
A quick look over Cauchy Theorem & Second Order Tensor special property
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 x1 (x), x2 (y) and x3 (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 dA1 is a projection of dAn on x2-x3 plane. Similarly, dA2 is a projection of dAn on x1-x3 plane and dA3 is a projection of dAn on x1-x2 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.
Cauchy Theorem relates Traction vector on a surface having arbitrary normal with the Stress tensor vector on a surface that has normal to a reference known (Cartesian) plane.
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.
Finally arriving at Navier-Cauchy Equation or Navier-Equation of Equilibrium
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 theorem.
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.
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 IInd part of this blog 😉