Lecture Notes on Pressure Effects

2.016 Hydrodynamics Reading #2 2.016 Hydrodynamics Prof. A.H. Techet Pressure effects Fluid forces can arise due to flow stresses (pressure and viscous shear), gravity forces, fluid acceleration, or other body forces. For now, let us consider a fluid in static equilibrium – with no velocity gradients (and thus, no viscous stresses). Forces on fluid in static equilibrium are due only to: 1. Pressure acting on the fluid volume 2. Gravity acting on the mass of the fluid 3. Other external body forces (e.g. electromagnetic) Using standard conventions, we consider pressure to be positive for compression. Recall that we said pressure is isotropic: it acts in all directions equally, and it always acts normal to surfaces. Let us prove that it acts in all directions equally by considering a triangular volume of fluid with height, dz, length, dx, and unit width, b, into the page: Figure 2.1: Elemental fluid volume version 4.0 updated 8/30/2005 -1- ©2005 A. Techet 2.016 Hydrodynamics Reading #2 The fluid element cannot support shear while at rest (by our definition of a fluid). Thus the sum of the forces on the triangle, in the x- and z- directions, MUST equal zero: ∑F x = px b(∆z ) − pnb(∆s) sin θ = 0 ∑ F = p (b∆x) −p (b∆s)cosθ − z z n 1 2 (2.1) ρ (b∆x∆z) = 0 (2.2) By geometry, ∆z = ∆s sin θ and ∆x = ∆s cosθ , such that ∑F = p z −pn = 0 (2.3) −pn − 12 ρ∆z = 0 . (2.4) x and ∑F = p z z Taking the limit as ∆x, ∆z goes to zero (i.e. the triangle goes to a point), we see that p x = p z = pn = p . (2.5) Since θ is arbitrary, pressure at a point in a fluid is independent of orientation and is thus isotropic. Pressure (or any stress for that matter) causes NO net force on a fluid element unless it varies spatially! version 4.0 updated 8/30/2005 -2- ©2005 A. Techet 2.016 Hydrodynamics Reading #2 Let us find how spatial pressure variation causes net force by considering a small fluid element: Figure 2.2: Fluid Element Volume (z is positive upwards by convention). Let us assume that the only forces are due to gravity and pressure gradients. Let the pressure on the bottom equal a reference pressure pbot = po . Then, the pressure on the top may be found using a Taylor’s series expansion: ptop = p0 + dp 1 d2 p (δz) + (δz)2 + ... dz 0 2! dz 2 0 (2.6) Taking the limit as δz goes to zero we are able to ignore the higher order terms and keep only up to second order. The resultant force due to pressure on the face is Fpress = ( pbot − ptop )dxdy = − dp dxdydz . dz 0 (2.7) We have defined Fpress to act in the positive z-direction. (We will resolve the negative sign when we sum forces.) Acting in tandem to the pressure force is the force due to gravity. The fluid density and volume give the mass of the element. Let us assume that our fluid element is small enough that the density is constant within the element. Therefore (defining the force in the positive z-direction, we have) Fweight = −mg = −ρVg version 4.0 updated 8/30/2005 -3- (2.8) ©2005 A. Techet 2.016 Hydrodynamics Reading #2 For static equilibrium, pressure forces MUST balance gravitational/body forces: − Fpress + Fweight = 0 (2.9) dp (dxdydz) − ρg(dxdydz) = 0 dz o (2.10) We can drop the subscript “o” from the pressure gradient term since we have taken the limit to a small element of fluid. This equation is valid at any point in the fluid. Canceling the elemental volume terms we are left with the hydrostatic equation. dp = − ρg dz Hydrostatic Equation: (2.11) The negative sign is valid here since the z-direction is taken positive pointing up, in the opposite direction to the gravitational force. In other words, the pressure decreases as your elevation increases. Remember, there is no pressure in outer space! Conversely, the pressure increases as your elevation decreases. The pressure at the bottom of the pool is higher than it is at the surface, which is why you have to equalize the pressure in your ears! (If we defined the z-direction to coincide with gravity then the negative sign would be dropped.) In this case, we only considered pressure variations in the z-direction (k̂) . In general, there are pressure changes in the x- or y-directions as well, and we define the pressure gradient as: ∇p = dp dp ˆ dp i+ ĵ + k̂ . dy dz dx (2.12) We can rewrite the hydrostatic equation as: ∇p = −ρgk̂ (2.13) The pressure gradient is always balanced by gravity, acceleration, viscous forces, and any other external body forces. version 4.0 updated 8/30/2005 -4- ©2005 A. Techet 2.016 Hydrodynamics Reading #2 Gauge and vacuum pressure: Pressure is usually referred to in one of two ways: 1) Absolute, or total, pressure 2) Relative to ambient (atmospheric) pressure a) p > pa Gage Pressure b) p < pa Vacuum pressure p(gage) = p – pa p(vacuum) = pa – p Since most pressure instruments measure the pressure in the fluid relative to atmospheric pressure, relative pressure is the commonly used quantity. In order to get the total (absolute) pressure from the gage or vacuum pressure, the atmospheric pressure must be known. P (Pascals) 120,000 High Pressure 30,000 90,000 Local Atmospheric 40,000 50,000 Vacuum Pressure 50,000 0 Absolute Zero tension Figure 2.3: Relative pressure chart version 4.0 updated 8/30/2005 -5- ©2005 A. Techet 2.016 Hydrodynamics Reading #2 Hydrostatic Force on a Wall: Recall the equation for the hydrostatic pressure gradient in a liquid is ∇p = −ρgk̂ (2.14) which, in the vertical direction this translates to or dp = − ρg , dz (2.15) dp = − ρg dz . (2.16) Integrating in the z-direction we get pressure as a function of depth: ∫ pa p dp = − ∫ z ρg dz 0 (2.17) pa − p = −ρg (0 − z) = ρgz (2.18) p = pa − ρgz (2.19) Note that z is defined as zero at the surface, so pressure increases with depth (since z < 0 under water) with a constant slope, ρ g . Pressure is either considered relative to a reference pressure or in absolute terms. Most pressure gauges are differential measurement devices that measure pressure relative to ambient (or atmospheric) pressure. Thus it is important to keep in mind the effects due to atmospheric pressure in your laboratories and calculations. Pressure is isotropic, and therefore pressure is the same on a vertical or horizontal surface at a given depth. Take for instance the ocean bottom and a vertical seawall that extends to the bottom. At depth H, the horizontal pressure acting on the wall is equivalent to that pressure acting along the entire seafloor at that same depth. Figure 2.4: Absolute Pressure vs. Gauge pressure version 4.0 updated 8/30/2005 -6- ©2005 A. Techet 2.016 Hydrodynamics Reading #2 Horizontal Force acting on a vertical surface: Using absolute pressure formulation we can find the force on the wall from the pressure as a function of depth: p = pa − ρgz (2.20) Thus the elemental force acting in the x-direction due to the pressure is dFx = p dA = ( pa − ρgz)w dz , (2.21) where w is the width of the wall into the page. To determine the force on the wall we must consider the pressure acting on both sides of the wall. Let’s assume that on the left of the wall water of depth, h, is exerting force F1 on the wall. The right side of the wall is open to the air with atmospheric pressure acting over the height of the wall. Under this setup we must account for the atmospheric pressure on both sides of the wall. The elemental force in the x-direction at some depth z below the free surface is formulated on the left and right sides of the wall to determine the total force acting on the wall: Left: dFl = ( pa − ρgz ) w dz (2.22) Right: dFr = − pa w dz (2.23) dFx = dFl + dFr = Pa w dz − ρgzw dz − Pa w dz (2.24) Total force: The two underlined terms cancel and we are left only with the effect due to the presence of the water. Integrating the pressure over the depth F= ∫ 0 −H −ρgzdz = 1 2 ρgwH 2 (2.25) If we use gauge pressure to calculate the force on the wall, we already take into account the effect of atmospheric pressure and can directly calculate the force from: pg = −ρgz (gauge pressure) version 4.0 updated 8/30/2005 -7- (2.26) ©2005 A. Techet 2.016 Hydrodynamics Reading #2 dF = pg dA = (−ρgz)w dz (2.27) Resulting in the same force found when we used absolute pressure. The force acting on the wall is a “distributed load” which acts along the entire depth of the wall. However it is possible to determine how this force “acts” on the wall using a simple moment balance: dMo = z × dF (2.28) where Mo is the moment on the wall about the origin and z is the moment arm perpendicular to the force direction. Figure 2.5: Moment balance on the wall. The elemental moment about the origin due to the hydrostatic pressure is: dMo = z ⋅ (−ρgz )wdz (2.29) Integrating leads to the total moment about the origin: Mo = − 1 3 ρgwH 3 version 4.0 updated 8/30/2005 -8- (2.30) ©2005 A. Techet 2.016 Hydrodynamics Reading #2 By definition the product of the force F times the moment arm z must equal the moment M: M =Fz (2.31) Such that the point at which the force acts on the wall is z : z= − 1 3 ρgwH 3 = −23 H 1 ρgwH 2 2 (2.32) Figure 2.6: The point at which the resultant force acts is known as the center of pressure. We can use a similar approach to the problem of pressure on a sloped wall. This is left for a homework exercise. Since atmospheric pressure acts everywhere then gage pressure is the ideal pressure to use in these exercises. The resultant forces and moments on the sloped wall can be found using simple geometry and then extended to the case of a “V”shaped ship hull. Figure 2.7: V-shaped Hull version 4.0 updated 8/30/2005 -9- ©2005 A. Techet