The Euler equations - Fluid Dynamics

1 Chapter 11 The Euler Acceleration of a Fluid parcelIn order to write the equations of motion for a Fluid , we must place the ob-server in an inertial reference system. Then we have to applythe second principleof Dynamics (Newton s second law) to its parcels. This requires a suitable expres-sion for the acceleration of the parcels as a function of the velocity field. Bearin mind that when we speak of the acceleration of a parcel, we mean the timederivative of its velocity. But, according to the Eulerian approach, we need toexpress this quantity by means of a velocity field defined as a function of thetime and space are in the same situation encountered in section [ ] forthe evaluation ofthe time variation of a scalar property following the motionof the parcel.

2 Thisrelation was extended to an arbitrary vector quantity in ( ). The only differenceis that now the velocity field that advects the property and the property itselfcoincide. Thus, in this case ( ) becomesdudt= u t+ (u )u.( )It should be noted that the time derivative on the left is atotal derivative, and7374 Franco Mattioli (University of Bologna)represents the acceleration of the parcel, while the time derivative on the rightis apartial derivative, that is, it represents the rate of change of the velocity infixed points of us analyze in greater detail the above expression. The acceleration of aparcel depends on two factors.

3 The former, u/ t, also calledlocal acceleration,represents the acceleration due to the fact that in a given point of space thevelocity of the parcels passing through it can be either increasing or decreasingin time. The latter, (u )u, also calledadvective term of the acceleration,provides the acceleration due to the fact that a parcel can move from a region oflow velocity to one of high velocity, or vice a unidirectional flow that is uniform over a horizontal plane and possibly variablewith the vertical coordinate, the advective component of the acceleration vanishes. Infact, the parcels during their motion do not undergo any change of us see how this statement can be translated in mathematical terms.

4 Letxbe thedirection of the motion. The three components of the advective term in thex-directionu u x+v u y+w u zall vanish: the first one vanishes because the velocityudoes not vary withx, and theother two because bothvandware zero. Furthermore, the other two components ofthe acceleration in theyandz-directions vanish because bothvandware the flow is also stationary, the local acceleration is zeroas well, so that the totalderivative of the velocity vanishes, in agreement with the fact that the parcels are second term of the acceleration is the source of the greatest mathematicaldifficulties encountered in the study of Fluid mechanics.

5 It is, in fact, anonlinearterm, indeed, a strongly nonlinear term. This prevents the use ofthe manymathematical tools available for the linear equations of all types, and forces theadoption and continuous research of new and increasingly advanced rectilinear pipe of variable circular sectionS=S(x) has a volume rateof flowQ. Evaluate the advective acceleration of the parcels lying along its central axis,assuming that their velocity is equal to the average velocity computed along a transversalsection of the that ifu= r, then(u )u= the physical meaning of such an of Fluid Dynamics ( ) us orientate the coordinate axes in such a way that = k.

6 Developingthe given expression by components, we obtain(u )( r) =(u x+v y)( yi+ xj) = vi+ uj= last term is nothing but the centripetal acceleration associated to the circular pathof the parcels. In fact, if we decomposerin the sum of the two componentsr andr ,respectively, parallel and perpendicular to the vector , we have u= ( r) = k [ k (r +r )] = 2r ,Problem we reverse the flows of the previous examples, does the signof theadvective component of the acceleration change?Problem the basis of the examples presented, we see that for a general motionthe advective acceleration is formed by at least two components: a tangential accelerationin the direction tangent to the trajectory of the parcel, anda centripetal accelerationin a direction normal to it, related to the curvature of the trajectory.

7 Are there otherterms in addition to these?Now that we have an expression for the parcel acceleration, we can write themomentum equation, , the equation which provides an expression of the parcelacceleration. By applying the second principle of dynamicsone has dudt x y z= F x y z.( )Here, is the Fluid density, x y zthe volume of the Fluid parcel, m= x y zits mass andFthe force per unit mass exerted on gravity force is clearly unaltered by the motion. Thus our attention mustbe concentrated on the surface stresses. We have already studied the behavior ofthe pressure necessary for an hydrostatic equilibrium.

8 Anychange in the valuesof the pressure with respect to its hydrostatic distribution will give rise to somekind of The Euler equationsThe only forces present in a Fluid at rest are the pressure forces. It is difficultto think that, as soon as the Fluid moves, such forces disappear or are deeplychanged. Thus, it seems reasonable to assume that the structure of the forces76 Franco Mattioli (University of Bologna)on the infinitesimal parcel remains the same found in the hydrostatic case alsofor a Fluid in motion. In other words, we may assume that the stresses betweenparcels are always normal to their surfaces of separation and independent oftheir orientation.

9 The symbol used to denote the scalar quantity defining themagnitude of this forces will remain the same, along with thename, pressure (ormore exactlydynamic pressure, to point out that now the pressure varies notonly in space, but also in time).In this way, the expression of the force would remain the samealready foundin ( ) as a function of the hydrostatic pressureF= p + inserting such expression in ( ) along with expression ( ) for the accel-eration of the parcel, we have u t+u u= p +g.( )These equations are known as theEuler equations , after the name of the authorthat derived them for the first time. Written in component form, they become u t+u u x+v u y+w u z= 1 p x,( ) v t+u v x+v v y+w v z= 1 p y,( ) w t+u w x+v w y+w w z= 1 p z g.

10 ( )Problem the condition for the validity of the hydrostatic 0, , when the vertical acceleration vanishes, ( ) reducesto ( ). In particular, the hydrostatic balance holds forany purely horizontal turns out that these equations can describe the structureof the motiononly in some circumstances. In many cases the solution is a good approximationof the real flow only in certain regions of space, but not in others. In other casesthe real motion is completely simplest hypothesis the we can formulate is that the pressure forces arenot the only forces present in a Fluid . Thus, other forces mustexist that certainkinds of motions might of Fluid Dynamics ( )