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I am very interested in the mathematical description of the next topics:

 1. Bernoulli equation
 2. Transition from laminar to turbulent flow

Could you help me by providing any documentation that you consider useful?

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Bernoulli Principle

The Bernoulli theorem is an extension of the conservation of energy, as are many theorems in the realm of physics. It states simply that the total pressure energy in a system is conserved. The Bernoulli principle is actually an approximation that is only valid for incompressible, one dimensional, isentropic flow. Meaning that the density of the fluid must be constant, the flow velocity does not vary in any direction but one, and there are no losses in the system (heat transfer, friction, etc). In aerodynamics, the bernoulli principle is only valid for flow velocities less than Mach 0.3, after which the results begin to diverge from reality by greater than 5%. In many hydrodynamic applications, the principle is acceptable to use in many applications; and is used frequently in civil engineering. The theorem is given mathematically by:

$$ P + \frac12\rho v^2 + \gamma z = constant $$ $$ P = pressure$$ $$ \rho = density$$ $$ v = velocity $$ $$ \gamma = \frac\rho g$$ $$ z = elevation $$

This formulation is especially useful for calculating flow parameters between two points, where some parameters are known. Because the pressure energy is conserved, we can equation the total pressure between two flow points:

$$ P_1 + \frac12\rho v_1^2 + \gamma z_1 = P_2 + \frac12\rho v_2^2 + \gamma z_2 $$

Which can be used to solve for flow pressure, velocity, cross sectional area, and elevation.

Turbulent Transition

The transition between laminar and turbulent flow regimes is a more difficult topic. The Reynolds number, given by:

$$ Re = {\rho \bar U d_h\over\mu}$$

Is a dimensionless parameter that describes the balance between inertial forces, and viscous forces in the flow. The top part of the fraction is the density, mean velocity, and hydraulic diameter multiplied, which represents inertial forces in the fluid. The bottom part of the fraction is the fluid viscosity, representing the viscous forces in the fluid. Therefore, at low Reynolds numbers, viscous forces are dominating the flow field resulting in a slow moving, predictable velocity distribution. At high Reynolds numbers, inertial forces begin to dictate the flow field and each fluid particle may change course erratically based on its inertia. It is at this point that the fluid may become turbulent, because we can no longer predict, with utmost certainty, what the flow field will look like. Different flow situations have different turbulent transition Reynolds numbers, but local turbulence can occur even at low Reynolds numbers depending on the properties of the fluid and flow conditions. Scientists have developed many ways of modelling turbulent behavior, which are beyond the scope of the question.

If these two questions are at all related, you should know that the Bernoulli formula is applicable (for the most part) for low Re numbers. At high Re, the frictional losses and viscous losses in the fluid are not ignorable, and we can no longer make this assumption.

For more information about turbulent transition: https://en.wikipedia.org/wiki/Laminar%E2%80%93turbulent_transition

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