FLUID DYNAMICSFLUID DYNAMICS

Phys 5306Phys 5306By Mihaela-Maria TanasescuBy Mihaela-Maria Tanasescu

GOVERNING EQUATIONSGOVERNING EQUATIONS

COANDA EFFECTCOANDA EFFECT

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

                  

                   

• The continuity assumptionThe continuity assumption::Knudsen NumberKnudsen NumberContinuum mechanicsContinuum mechanicsModeling fluidsModeling fluids

• Governing equationsGoverning equationsConservation equationsConservation equationsConstitutive equationsConstitutive equations

• Aerodynamics applicationAerodynamics applicationPhysics of flight and the Coanda Physics of flight and the Coanda

effecteffect

Knudsen numberKnudsen number

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

The continuity assumptionThe continuity assumption

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

• density ρ(r,t)

• flow velocity u(r,t)

• pressure p(r,t)

• temperature T(r,t)

The continuum method is The continuum method is generaly used to describe fluid generaly used to describe fluid dynamicsdynamics• The vast majority of phenomena The vast majority of phenomena

encoutered in fluid mechanics fall encoutered in fluid mechanics fall well within the continuum domain well within the continuum domain and may involve liquids as well as and may involve liquids as well as gasesgases

Modeling fluidsModeling fluids

• Eulerian descriptionEulerian description: a fixed reference : a fixed reference frame is employed relative to which a fluid frame is employed relative to which a fluid is in motion;is in motion;

• Time and spatial position in this reference frame, Time and spatial position in this reference frame, {t, r}{t, r} are used as are used as independent variablesindependent variables

• The fluid variables such as mass, density, The fluid variables such as mass, density, pressure and flow velocity which describe the pressure and flow velocity which describe the physical state of the fluid flow in question are physical state of the fluid flow in question are dependent variablesdependent variables as they are functions as they are functions of of the independent variables the independent variables

Modeling fluidsModeling fluids

• Lagrangian descriptionLagrangian description the fluid the fluid is described in terms of its is described in terms of its constituent fluid elements;constituent fluid elements;

• Attention is fixed on a particular Attention is fixed on a particular mass of fluid as it flowsmass of fluid as it flows

Modeling fluids

• Control volumes• The control volume is arbitrary in shape and each conservation

principle is applied to an integral over the control volume

Modeling fluidsModeling fluids

• Reynold’s Transport Theorem:Reynold’s Transport Theorem:• Relates the lagrangian derivative of a volume Relates the lagrangian derivative of a volume

integral of a given mass to a volume integral in integral of a given mass to a volume integral in which the integrand has eulerian derivatives onlywhich the integrand has eulerian derivatives only

( )V V

DdV u dV

Dt t

Governing equations Governing equations

• The governing equations consist of The governing equations consist of conservation equations and conservation equations and constitutive equations; constitutive equations;

• conservation equations apply conservation equations apply whatever the material studied; whatever the material studied;

• constitutive equations depend from constitutive equations depend from the material; the material;

Governing equationsGoverning equations Conservation equationsConservation equations• Conservation of Conservation of

mass-Continuity mass-Continuity equation:equation:

• Continuity equation Continuity equation for an for an incompressible fluid:incompressible fluid:

( ) 0kku

t x

0k

k

u

t x

Governing equationsGoverning equations Conservation equationsConservation equations

• Conservation of momentumConservation of momentum

The principle of conservation of The principle of conservation of momentum is in fact an application momentum is in fact an application of Newton’s second law of motion to of Newton’s second law of motion to an element of fluid an element of fluid

j j ijk i

k i

u uu f

t x x

Governing equationsGoverning equations Conservation equationsConservation equations

• Conservation of energyConservation of energy the modified form of the first law of the modified form of the first law of thermodynamics applied to an element of fluid thermodynamics applied to an element of fluid states that the rate of change in the total energy states that the rate of change in the total energy (intrinsic plus kinetic) of the fluid as it flows is (intrinsic plus kinetic) of the fluid as it flows is equal to the sum of the rate at which work is equal to the sum of the rate at which work is being done on the fluid by external forces and the being done on the fluid by external forces and the rate on which heat is being added by conduction rate on which heat is being added by conduction

2k k j i j

kk k k i j jj j

e e u T u u u uu p k

t x x x x x x x x

Constitutive equationsConstitutive equations  • The nine elements of the stress tensor have been The nine elements of the stress tensor have been

expressed in terms of the pressure and the velocity expressed in terms of the pressure and the velocity gradients and two coefficients gradients and two coefficients and and .. These coefficients These coefficients cannot be determined analytically and must be determined cannot be determined analytically and must be determined empirically. They are the empirically. They are the viscosity coefficientsviscosity coefficients of the of the fluid. fluid.

• The second constitutive relation is Fourier’s Law for heat The second constitutive relation is Fourier’s Law for heat conduction conduction

k j iij ij ij

k i j

u u up

x x x

jj

Tq k

x

Navier-Stokes Equations Navier-Stokes Equations   

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

  

Navier-Stokes EquationsNavier-Stokes Equations

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

Navier-Stokes Equations Navier-Stokes Equations   

j j k i jk

k j j k i j i

u u p u u uu f

t x x x x x x x

2

2

j j jk i

k j i

u u p uu f

t x x x

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

• Coanda EffectCoanda Effect

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

as air at usual temperature, pressures and speedsas air at usual temperature, pressures and speeds

Henri CoandaHenri Coanda

• Romanian Scientist (1886-1972)Romanian Scientist (1886-1972)

• One of the pioneers of the aviation, parent of the modern One of the pioneers of the aviation, parent of the modern jet aircraftjet aircraft

• Coanda-1910Coanda-1910 - a revolutionary aircraft in many ways. First - a revolutionary aircraft in many ways. First and foremost, it is now being recognized as the first jet and foremost, it is now being recognized as the first jet engine aircraft, making its engine aircraft, making its first and only flight on 16 first and only flight on 16 December, 1910December, 1910. Coanda's aircraft was the first to have . Coanda's aircraft was the first to have no propeller. This was 30 years prior to Heinkel, Campini, no propeller. This was 30 years prior to Heinkel, Campini, and Whittle who have been considered the "fathers" of jet and Whittle who have been considered the "fathers" of jet flight. Missing financial support, Coanda did not pursue flight. Missing financial support, Coanda did not pursue further development of his "reactive" aircraft further development of his "reactive" aircraft

• The engine was the real innovationThe engine was the real innovation, and it is lost to the , and it is lost to the aircraft industry that development was not further pursued aircraft industry that development was not further pursued in 1910. in 1910.

Henri CoandaHenri Coanda

• Aerodina lenticulara Aerodina lenticulara in 1934 he was granted in 1934 he was granted a French patent related to the a French patent related to the Coandă EffectCoandă Effect; ;

• in 1935, he used the same principle as the basis for a in 1935, he used the same principle as the basis for a hovercraft called "Aerodina Lenticulara", which was hovercraft called "Aerodina Lenticulara", which was very similar in shape to the flying saucers;very similar in shape to the flying saucers;

• later being bought by USAF and become a later being bought by USAF and become a classified projectclassified project

• Henri Coanda’s sketches for his “aerodina Henri Coanda’s sketches for his “aerodina lenticulara”lenticulara”

                

   

                

   

                   

                

   

            

              

              

              

 

Henri CoandaHenri Coanda

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