Fluid Mechanics, Lecture Notes- Physics - 2, Study notes of Physics

Download Fluid Mechanics, Lecture Notes- Physics - 2 and more Study notes Physics in PDF only on Docsity! Engineering Tripos 1B Paper 4 Fluid Mechanics Lecture 2 • Changes due to motion through a field; • Newton’s second law (f = ma) applied to a fluid: Euler’s equation; • Euler’s equation integrated along a streamline: Bernoulli’s equation; • Bernoulli’s equation and streamline curvature; • Determining the pressure field from a flow’s streamlines. 1B Thermofluids website The 1B Thermofluids website contains extra explanations of the key concepts in the course and some worked examples: • The address is: http://camtools.caret.cam.ac.uk/ • The course name is: Thermofluids : EngIB • A Raven login is required. Matthew Juniper mpj1001@cam.ac.uk 1 2.1 Changes due to motion through a field The contour lines on this map of Snowdonia show average height above sea level. At every point (x, y) this height field has a single value of h(x, y). As you walk along a path at a certain velocity v at what rate does your height change? The rate of change of your height is given by v ·∇h. We can write this as v · (∇h) or as (v · ∇)h. These are equivalent but the second version becomes more convenient later because (v · ∇) is a scalar operator that acts on anything to its right. In this case it acts on h but later it will act on other variables. (v · ∇) =     vx vy vz   ·   ∂/∂x ∂/∂y ∂/∂z     = ( vx ∂ ∂x + vy ∂ ∂y + vz ∂ ∂z ) 2 2.2 Newton’s second law: f = ma If we put a neutrally buoyant solid cube into a fluid flow and ignore all the viscous forces, we can work out the net force on the cube by considering the pressure on each face: f =   fx fy fz   = −   ∂p/∂x ∂p/∂y ∂p/∂z   δxδyδz = − ∇p δxδyδz Now we write f = ma for the cube: We could do this for an imaginary cube (or blob) of fluid instead. Similarly, we would find that the cube (or blob) of fluid is being accelerated or decelerated by the pressure gradients in the flow. However, we know how to express the acceleration of a fluid blob in terms of the fluid’s velocity field because we worked it out at the end of the previous section. We can put them together to get: This is the Euler equation, which becomes the Navier-Stokes equation when viscous terms are included (see p9 of the databook, where it is called by its other name: the momentum equation). The Euler equation is simply f = ma applied to an inviscid fluid. Remember that there is one equation for each dimension. For example in 3D cartesian coordinates there is one equation in the x-direction, one in the y-direction and one in the z-direction. 5 Link to 1st year Momentum Equation In the 1st year you derived the Steady Flow Momentum Equation (SFME) and the more general Momentum Equation (p7 of databook) by considering a control volume: d dt ∫ V ol ρv dV + ∫ Surf ρv(v · dA) = F − ∫ Surf p dA The Euler equation is the same thing, but in vector notation: ∂ρv ∂t + ρ(v · ∇)v = body forces − ∇p 6 2.3 Euler’s equation applied along a straight streamline We will look at the central streamline of the steady flow around a cylinder. The fluid on this streamline flows at velocity V in the x-direction. This streamline is easy to examine because the streamline coordinate system is aligned with the cartesian coordinate system. The more general case is considered in the next section. The cartesian unit vectors are (ex, ey) and ∇ is defined as ∇ ≡ ex∂/∂ x + ey∂/∂y. Euler’s equation (for an incompressible fluid) is: ρ ∂v ∂t + ρ(v · ∇)v = −∇p but the y-velocity is zero and the flow is steady, so it becomes: Re-arranging the ex component gives: ∂ ∂x ( 1 2 ρV 2 ) + ∂p ∂x = 0 ∂ ∂x ( p + 1 2 ρV 2 ) = 0 and this can be integrated along the streamline to give: This is Bernoulli’s equation. It has arisen naturally from f = ma and the assumption of no viscous forces. When we integrated it along the streamline, the force terms became energy terms in exactly the same way that the change in potential energy of a mass in a gravitational field is equal to the force integrated over the distance it moves. Indeed, the calculation can easily be repeated with gravity, which causes an extra ρgh term. So each term can be thought of as an energy per unit volume. Bernoulli’s equation applies along any streamline, as we show in the next section. Furthermore, if there is no vorticity in the flow then every streamline has the same Bernoulli constant. 7