FLUID DYNAMICSPhys 5306 By Mihaela-Maria Tanasescu Mihaela-

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. systems.

The continuity assumption: assumption:Knudsen Number Continuum mechanics Modeling fluids

Governing equationsConservation equations Constitutive equations

Aerodynamics applicationPhysics of flight and the Coanda effect

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

The continuity assumptionThe 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 wellassumed to vary continuously from one point to another. The discrete, molecular nature of a fluid is ignoreddensity (r,t) flow velocity u(r,t) pressure p(r,t) temperature T(r,t)

The continuum method is generaly used to describe fluid dynamicsThe vast majority of phenomena encoutered in fluid mechanics fall well within the continuum domain and may involve liquids as well as gases

Modeling fluidsEulerian description: a fixed reference frame is description: 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 fluidsLagrangian description the fluid is described in terms of its constituent fluid elements; Attention is fixed on a particular mass 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 fluidsReynold s Transport Theorem: 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 xE V E dV ! V xt y (E u) dV Dt

Governing equationsThe governing equations consist of conservation equations and constitutive equations; conservation equations apply whatever the material studied; constitutive equations depend from the material;

Governing equationsConservation equationsConservation of massmassContinuity equation:

xV x ( V uk ) ! 0 xt xxk

Continuity equation for an incompressible fluid:

xV xuk V !0 xt xxk

Governing equationsConservation 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

xuj xuj xW ij V V uk ! V fi xt xxk xxi

Governing equationsConservation equationsConservation 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 conductionxe xe xuk x xT ! p V V uk k xt xxk xxk xx j xx j2 xuk xuj xui xuj P Q xxk xxi xxj xxj

Constitutive equationsThe nine elements of the stress tensor have been expressed in terms of the pressure and the velocity gradients and two coefficients P and Q. These coefficients cannot be determined analytically and must be determined empirically. They are the viscosity coefficients of the fluid.

xuk xuj xui W ij ! pH ij PH ij Q xxk xxi xxj The second constitutive relation is Fourier s Law for heat conduction

xT qj ! k xxj

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

NavierNavier-Stokes EquationsThe central equations for fluid dynamics are the NavierNavierStokes equations, which are non-linear differential nonequations that describe the flow of a fluid whose stress depends linearly on velocity and on pressure. The unsimplified equations do not have a general closedclosedform 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

NavierNavier-Stokes Equationsxuj xuj xp x xuk x xui xuj V V uk ! Q P V f xt xxk xxj xxj xxk xxi xxj xxi

xuj xuj xp x uj V V uk ! Q 2 V fi xt xxk xxj x xi2

"That we have written an equation does notremove from the flow of fluids its charm or mystery or its surprise." --Richard Feynman --Richard [1964]

Coanda Effect

"That we have written an equation does notremove from the flow of fluids its charm or mystery or its surprise." --Richard Feynman --Richard [1964]

Coanda effectThe Coanda Effect works with any of our usual fluids, such as air at usual temperature, pressures and speeds

Henri CoandaRomanian Scientist (1886-1972) (1886One of the pioneers of the aviation, parent of the modern jet aircraft CoandaCoanda-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. 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 innovation, industry that development was not further pursued in 1910.

Henri Coandain 1934 he was granted a French patent related to the Coand Effect; 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

Aerodina lenticulara

Henri Coanda"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."