MrsDrPoe: Conservation of Momentum, Part V

It's Thesis Tuesday on the blog, and today we'll be continuing our examination of the Conservation of Momentum equation, this time focusing on the inviscid reduction of its differential form.

If you recall, the difficulty with the differential momentum equation is that we need to define the stress terms. Previously, we obtained the following results:

d/dx(Sxx) + d/dy(Txy) + d/dz(Txz) = -d/dx(p) + mu*(d/dx(d/dx(u))+d/dy(d/dy(u)) + d/dz(d/dz(u)))

d/dx(Tyx) + d/dy(Syy) + d/dz(Tyz) = -d/dy(p) + mu*(d/dx(d/dx(v))+d/dy(d/dy(v)) + d/dz(d/dz(v)))

d/dx(Tzx) + d/dy(Tzy) + d/dz(Szz) = -d/dz(p) + mu*(d/dx(d/dx(w))+d/dy(d/dy(w)) + d/dz(d/dz(w)))

If we examine the case of inviscid flow, this means we are looking at an area where viscous effects are negligible (mu = 0). Generally this occurs in the far-field, i.e. in a region of the fluid far away from any walls or boundaries. This leaves us with:

d/dx(Sxx) + d/dy(Txy) + d/dz(Txz) = -d/dx(p)

d/dx(Tyx) + d/dy(Syy) + d/dz(Tyz) = -d/dy(p)

d/dx(Tzx) + d/dy(Tzy) + d/dz(Szz) = -d/dz(p)

So our stress tensor terms are ONLY related to changes in pressure. If we replace these new equalities in the differential form of the equation, we end up with a set of equations known as Euler's Equations. It is important to note, however, that this set is only valid for inviscid flows.

If we apply the assumptions of steady, incompressible flow with no outside forces acting on the fluid to Euler's Equations, we get:

u*d/dx(u) + v*d/dy(u) + w*d/dz(u) = gx - (1/rho)*d/dx(p)


u*d/dx(v) + v*d/dy(v) + w*d/dz(v) = gy - (1/rho)*d/dy(p)


u*d/dx(w) + v*d/dy(w) + w*d/dz(w) = gz - (1/rho)*d/dz(p)

Using a vector identity (which we won't discuss in further detail), and applying along a streamline (lines always parallel or tangent to the flow), we can further reduce this equation to:

0.5*V*V + (p/rho) + g*y = constant (along a streamline)

or: 0.5*V1*V1 + (p1/rho) + g*y1 = 0.5*V2*V2 + (p2/rho) + g*y2

This is Bernoulli's Equation, and it is written for two specific points along a streamline.

Restrictions:

Remember that this equation is only applicable to flows of fluids where the assumptions we made to derive the equation (steady, incompressible, inviscid, along a streamline, no outside forces) hold. It is CRUCIAL to realize that if the flow does not meet these qualifications, this simplified equation CANNOT be used!

If our flow cannot be treated as incompressible, but the other assumptions hold, we can modify the Bernoulli Equation so that it can still be used. For compressible gases, we can apply the ideal gas law to obtain:

0.5*V*V + y*g + R*T*ln(p) = constant (along a streamline)

For unsteady flows, we can modify the equation by the addition of a single term integrating the acceleration over a streamline:

0.5*rho*V1*V1 + p1 + rho*g*y1 = rho*int(d/dt(V))ds + 0.5*rho*V2*V2 + p2 + rho*g*y2

Physical Meaning:

The Bernoulli Equation is simply the mathematical statement of the principle "the work done on a particle by all forces acting on the particle is equal to the change of the kinetic energy of the particle." We can better see that if we look at each of the terms in the equation. As the particle moves, we see forces of pressure and gravity acting on it (the z*g and p/rho terms). This is directly connected to the amount of kinetic energy that the particle has (0.5*V*V).

If we multiply each term in the Bernoulli Equation by the density, rho:

0.5*rho*V*V + p + g*rho*y = constant (along a streamline)

we obtain the pressure form of the equation. Here p represents the thermodynamic pressure of the fluid as it flows (this pressure determines the state of a fluid, i.e. whether it is a gas or a liquid). This value is typically termed the static pressure of the fluid, since it could be measured by moving along with the fluid or being static in relation to the fluid. The third term is the hydrostatic pressure, which is associated with the hydrostatic pressure condition. The second term is called the dynamic pressure since it is associated with the velocity of the fluid.

The combination of the first two terms in the equation is the stagnation pressure. This pressure represents the conversion of all the kinetic energy into a pressure rise, which occurs at a stagnation point where the velocity is equal to zero. The streamline leading to a stagnation point is called a stagnation streamline. The sum of all three terms in this form of the Bernoulli Equation is known as the total pressure, which is constant along a streamline.

Often in Mechanical Engineering, the Bernoulli Equation will be written in head form; this form of the equation is particularly useful for pipe flow and can be derived by dividing each term in the original equation by the gravitational constant, g:

(p/gamma) + (0.5*V*V/g) + y = constant (along a streamline)

In this form, the pressure term is called the pressure head, which represents the height of a column of fluid needed to produce the pressure p. The velocity term is called the velocity head, which represents the vertical distance needed for the fluid to fall freely to reach the velocity V from rest. The height term is the elevation head, and it represents the potential energy of the fluid.


Next week we'll look at some very simple examples of how to employ the Bernoulli Equation. Stay tuned!