Phys 5306
By Mihaela-Maria Tanasescu
GOVERNING EQUATIONS
COANDA EFFECT

• Fluid dynamics is the key to our understanding of some of the most important phenomena in our physical world: ocean currents and weather systems .

•
:
Knudsen Number
Continuum mechanics
Modeling fluids
•
Conservation equations
Constitutive equations
•
Physics of flight and the Coanda effect

• Problems with Knudsen numbers at or above unity must be evaluated using statistical mechanics for reliable solutions

The continuity assumption
• The continuity assumption considers fluids to be continuos. That is, properties such as density, pressure, temperature, and velocity are taken to be well-defined at infinitely small points, and are assumed to vary continuously from one point to another. The discrete, molecular nature of a fluid is ignored
• density ρ(r,t)
• flow velocity u(r,t)
• pressure p(r,t)
• temperature T(r,t)

• The vast majority of phenomena encoutered in fluid mechanics fall well within the continuum domain and may involve liquids as well as gases

• Eulerian description : a fixed reference frame is employed relative to which a fluid is in motion;
• Time and spatial position in this reference frame, {t, r} are used as independent variables
• The fluid variables such as mass, density, pressure and flow velocity which describe the physical state of the fluid flow in question are dependent variables as they are functions of the independent variables

Modeling fluids
• Lagrangian description the fluid is described in terms of its constituent fluid elements;
• Attention is fixed on a particular mass of fluid as it flows

• Control volumes
• The control volume is arbitrary in shape and each conservation principle is applied to an integral over the control volume

Modeling fluids
• Reynold’s Transport Theorem:
• Relates the lagrangian derivative of a volume integral of a given mass to a volume integral in which the integrand has eulerian derivatives only
D
Dt

V
 dV
 
V


 
 t
  
(

)



• The governing equations consist of conservation equations and constitutive equations;
• conservation equations apply whatever the material studied;
• constitutive equations depend from the material;

Governing equations
Conservation equations
• Conservation of mass-
Continuity equation:





 t x k
(
 u k
)

0
• Continuity equation for an incompressible fluid:


  t
 u k
 x k

0

Governing equations
Conservation equations
• Conservation of momentum
The principle of conservation of momentum is in fact an application of Newton’s second law of motion to an element of fluid


 u t j   u k

 u x k j 


 x i ij   f i

Governing equations
• Conservation of energy the modified form of the first law of thermodynamics applied to an element of fluid states that the rate of change in the total energy (intrinsic plus kinetic) of the fluid as it flows is equal to the sum of the rate at which work is being done on the fluid by external forces and the rate on which heat is being added by conduction

 e
 t
  u k
 e
 x k
  p

 u k 

 x k x j

 k


T x j

  



 u x k k


2
 


 u x i j 

 u x j i


 u x j j

• The nine elements of the stress tensor have been expressed in terms of the pressure and the velocity gradients and two coefficients
 and  . These coefficients cannot be determined analytically and must be determined empirically. They are the viscosity coefficients of the fluid.
 ij
  p
 ij
  ij
 u k
 x k
 


 u x j i

 u x j
• The second constitutive relation is Fourier’s Law for heat conduction q j
  k

T
 x j

The equation of momentum conservation together with the constitutive relation for a
Newtonian fluid yield the famous Navier-Stokes equations, which are the principal conditions to be satisfied by a fluid as it flows

• The central equations for fluid dynamics are the Navier-
Stokes equations, which are non-linear differential equations that describe the flow of a fluid whose stress depends linearly on velocity and on pressure. The unsimplified equations do not have a general closedform solution, so they are only of use in computational fluid dynamics. The equations can be simplified in a number of ways. All of the simplifications make the equations easier to solve. Some of them allow appropriate fluid dynamics problems to be solved in closed form


 u j
 t
  u k
 u j
 x k
 

 p x j


 x j



 u k x k



 x i


 
 u x j i 

 u x i j



  f

 u j
 t
  u k
 u j
 x k
 

 p x j
 
 2 u j
 2 x i
  f i

That we have written an equation does not remove from the flow of fluids its charm or mystery or its surprise." --Richard Feynman
[1964]
• Coanda Effect

That we have written an equation does not remove from the flow of fluids its charm or mystery or its surprise." --Richard Feynman
[1964]
• Coanda effect
• The Coanda Effect works with any of our usual fluids, such as air at usual temperature, pressures and speeds

• Romanian Scientist (1886-1972)
• One of the pioneers of the aviation, parent of the modern jet aircraft
• Coanda-1910 - a revolutionary aircraft in many ways. First and foremost, it is now being recognized as the first jet engine aircraft, making its first and only flight on 16 December, 1910.
Coanda's aircraft was the first to have no propeller. This was 30 years prior to Heinkel, Campini, and Whittle who have been considered the "fathers" of jet flight. Missing financial support,
Coanda did not pursue further development of his "reactive" aircraft
• The engine was the real innovation, and it is lost to the aircraft industry that development was not further pursued in 1910.

• Aerodina lenticulara in 1934 he was granted a
French patent related to the Coandă Effect ;
• in 1935, he used the same principle as the basis for a hovercraft called "Aerodina Lenticulara", which was very similar in shape to the flying saucers;
• later being bought by USAF and become a classified project
• Henri Coanda’s sketches for his “aerodina lenticulara”

"These airplanes we have today are no more than a perfection of a child's toy made of paper. In my opinion, we should search for a completely different flying machine, based on other flying principles. I imagine a future aircraft, which will take off vertically, fly as usual, and land vertically. This flying machine should have no moving parts. This idea came from the huge power of cyclones."