AE301 Aerodynamics 1 (Hayashibara)
Understanding the Bernoulli’s Equation, learned in Fluid Mechanics
Bernoulli’s equation is such fundamental equation, yet do we really understand this equation? The following description is taken from NASA K-12 Education website of Glen Research Center:
In the 1700s, Daniel Bernoulli investigated the forces present in a moving fluid. There are of many forms of Bernoulli's equation. The equation appears in many physics, fluid mechanics, and airplane textbooks. The equation states that the static pressure p in the flow plus the dynamic pressure, one half of the density times the velocity V squared, is equal to a constant throughout the flow. We call this constant the total pressure pt of the flow.
Macro Scale Derivation (Continuity) of Bernoulli’s Equation
Thermodynamics is the branch of science which describes the macro scale properties of a fluid. One of the principle results of the study of thermodynamics is the conservation of energy; within a system, energy is neither created nor destroyed but may be converted from one form to another. We shall derive Bernoulli’s equation by starting with the conservation of energy equation. Assuming a steady, inviscid flow (also ignoring the gravity) we have a simplified conservation of energy equation (first law of thermodynamics) in terms of the total enthalpy of the fluid:
htt21 h q w where ht is the total enthalpy of the fluid (enthalpy + kinetic energy), q is the heat transfer into the fluid, and w is the useful work done by the fluid. Assuming no heat transfer into the fluid, and no work done by the fluid, we have:
hhtt21
From the definition of enthalpy (internal energy + flow energy) and total enthalpy (enthalpy + kinetic energy):
2222 22 VV21VV21 VV21 e pv e pv => hh => cpp T21 c T (!) wait . . . Isn’t this . . . 2121222122 22 where e is the internal energy, p is the pressure, v is the specific volume, and V is the velocity of the fluid. From the first law of thermodynamics if there is no work and no heat transfer, the internal energy remains the same:
VV22 pv 21 pv 2122
The specific volume is the inverse of the fluid density :
22 pp VV21 2122
Assuming that the flow is incompressible, the density is a constant. Multiplying the energy equation by the constant density:
VV22 pp21 (!) wait . . . I ended up in Bernoulli’s equation . . . 2122
This is the simplest form of Bernoulli's equation and the one most often quoted in textbooks. If we make different assumptions in the derivation, we can derive other forms of the equation.
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As it is clearly demonstrated, the Bernoulli’s equation can be derived BOTH from conservation of momentum and conservation of energy (traditionally, this is referred as very renowned the Bernoulli’s Principle of Conservation Laws).
• Bernoulli’s equation, in class, was derived from the conservation of linear momentum with assumptions of (i) steady flow, (ii) inviscid, and (iii) no body force (Euler’s equation) with limitation of (iv) M < 0.3 (incompressible subsonic). VV22 V 2 pp21 or p constant (along a given streamline) 2122 2
• However, an alternative form of Bernoulli’s equation (we called it, the energy equation) can be derived from the conservation of energy with assumptions of (i) steady, (ii) inviscid, (iii) no body force (Euler’s equation), (iv) use of the first law of thermodynamics (means: flow field is isentropic), and (v) constant specific heat (means: calorically perfect ideal gas). VV22 V 2 c T21 c T or cTconstant (along a given streamline) pp2122 p 2 Since a simple addition of incompressible (M < 0.3) flow will turn this equation into the familiar form of Bernoulli’s equation above, now you MUST realize that this is simply one of many variants of the Bernoulli’s principle based equations.
From Fluid Mechanics (read: Unit C-1 of Self-Study Materials of Fluid Mechanics), you should have known at least three different forms of Bernoulli’s equation:
The energy equation: or is, simply, the variant of Bernoulli’s equation (in conservation of energy form); what we did in class was simply we “removed” the constraint of constant density, or incompressible, limitation (as we do not want to be limited for M < 0.3 flows only) and “added” the constrains of isentropic flow and calorically perfect ideal gas.
(IMPORTANT) Understanding the equation derivation requires deeper understanding of basic physics of flow field. I expect you to focus on “where” “why” and “how” each equation is derived and used in aerodynamics.
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