Hydrostatic Pressure Effects and Archimedes’ Principle, Exercises of Fundamentals of Design

Download Hydrostatic Pressure Effects and Archimedes’ Principle and more Exercises Fundamentals of Design in PDF only on Docsity! 2.00AJ/16.00AJ Reading for Lecture #4 2.00AJ/16.00AJ Reading: Hydrostatic Pressure Effects and Archimedes’ Principle Pressure Effects I. Hydrostatic Pressure 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 (thus no viscous stresses). Forces are then due only to: 1. Pressure acting on the fluid volume 2. Gravity acting on the mass of the fluid 3. External body forces Using standard conventions we will consider pressure to be positive for compression. Recall that we said pressure is isotropic: Consider a triangular volume of fluid with height, dz, length, dx, and unit width, b, into the page: Figure 2.1: Elemental fluid volume -1- docsity.com 2.00AJ/16.00AJ Reading for Lecture #4 The fluid element can support no 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 = p x b(!z) " p n b(!s)sin#$ = 0 (2.1) F z = p z b(!x) " p n b(!s)cos#$ " 1 2 % b(!x!z) = 0 (2.2) We can simplify the above equations using simple geometry: !z = (!s)sin" and !x = (!s)cos" , (2.3) such that and F x = p x ! p n( )" = 0 F z = p z ! p n !1 / 2" (#z)$ = 0 (2.4) . (2.5) Taking the limit as ∆x, ∆z goes to zero (i.e. the triangle goes to a point), we see that p x = p z = p n = p . (2.6) 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! docsity.com 2.00AJ/16.00AJ Reading for Lecture #4 We can rewrite the hydrostatic equation as: ! !p = "# ! g (2.18) If there were additional gradients in the x- and y-directions similar steps could be followed and the resulting equation would be similar. Pressure gradient is always balanced by gravity, acceleration, viscous forces, and any other external body forces. II. 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 Since most pressure instruments are differential measurement devices, meaning they measure the pressure in the fluid relative to atmospheric pressure, absolute pressure is a commonly used quantity. Pressure is either greater or less than the ambient (atmospheric) pressure: 1) p > pa Gage Pressure p(gage) = p – pa 2) p < pa Vacuum pressure p(vacuum) = pa – p In order to get the total (absolute) pressure, the atmospheric pressure must be known. docsity.com 2.00AJ/16.00AJ Reading for Lecture #4 tension Figure 2.3: Relative pressure chart III. Hydrostatic Forces on a Wall Recall the equation for the pressure gradient in a liquid is P (Pascals) 120,000 90,000 50,000 0 50,000 40,000 30,000 High Pressure Local Atmospheric Vacuum Pressure Absolute Zero ! !p = "# ! g . (2.19) Which, in the vertical direction this translates to dp dz = !"g , (2.20) or dp = !"g dz . (2.21) Integrating in the z-direction we get pressure as a function of depth: dp p pa ! = " #g dzz H ! p ! p a = !"g z ! H( ) = "g H ! z( ) (2.22) (2.23) Note that pressure increases with depth with a constant slope, !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 docsity.com 2.00AJ/16.00AJ Reading for Lecture #4 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 depth h. 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. Horizontal Force acting on a vertical surface: Figure 2.4: Absolute Pressure vs. Gauge pressure Using absolute pressure formulation we can find the force on the wall from the pressure as a function of depth: p ! pa = !"g z ! h( ) (2.24) Thus the elemental force acting in the x-direction due to the pressure is dFx = pdA = pwdz , (2.25) 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 docsity.com 2.00AJ/16.00AJ Reading for Lecture #4 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 docsity.com 2.00AJ/16.00AJ Reading for Lecture #4 Archimedes’s Principle and Static Stability for Floating Vessels IV. Archimedes’s Principle: The force on a body due to pressure alone (in the absence of viscous forces) acts in the normal direction (recall pressure is isotropic!). The force can be calculated by integrating the pressure around the body: !!= S dsnpF ˆ ! where pressure is a function of depth below the free surface. Hydrostatic pressure is found using the equation: gzzp !"=)( . Thus the force due to pressure can be simplified as follows: !!!! "== SS dsnzgdsnpF ˆˆ # ! By calculus, the surface integral can be converted into a volume integral: ˆ S V z n ds z d= ! "## ### So that the force becomes: ˆˆ ˆ S S v F p nds g z nds g d g k! ! != = " = # = #$$ $$ $$$ ! We can see now that the buoyancy force acts to counterbalance the displaced volume ( � ) of fluid. For a semi-submerged body (partially submerged) the area of the water plane must be accounted for in the integration. V. Moment on a body (Ideal Fluid) The moment on a submerged body follows directly from structural mechanics or dynamics methodologies. ),,(),,()ˆ( 321 zyx S MMMMMMdsnxpM =!"= ## !! docsity.com 2.00AJ/16.00AJ Reading for Lecture #4 Recall from vector calculus: )(ˆ )(ˆ )(ˆˆˆˆ ˆ 12 13 23 321 nynxk nznxj nznyi nnn zyx kji nx ! ! ! + !==" ! So we get the moments as follows: !! "= s dsnznypM )( 231 about the x-axis We can also calculate in a similar fashion: M2 about the y-axis and M3 about the z- axis. VI. Center of Buoyancy Calculating the Center of buoyancy, it is first necessary to find the center of area. Area: != dxyA docsity.com 2.00AJ/16.00AJ Reading for Lecture #4 IX. Small rotation about the y-axis equation for stability: Sum of the torques about the center of rotation must be zero (conservation of angular momentum!) We can write the equation of motion in terms of the rotational displacement, θ, and rotational acceleration, !!! : 0I l! !+ =!! where l = |GB|. This is the equation of simple harmonic motion. The motion is in stable equilibrium when l >0 and unstable when l < 0. (You’ll see this later in dynamics class!) X. Metacenter: A ships metacenter is the intersection of two lines of action of the buoyancy force. What does this mean? Well, basically it means that as a ship rolls (heels) through an angle, θ, the center of buoyancy, B, shifts along a semi-elliptical path (depending on geometry). Since the buoyancy always acts vertically, the two lines of action for two different heel angles must cross at a point. This point is the metacenter of the ship, denoted by M in the figure below. docsity.com 2.00AJ/16.00AJ Reading for Lecture #4 The metacenter is the intersection of two distinct lines of action of the buoyancy force. When on at an angle of roll (heel) the center of buoyancy and center of gravity no longer act on the same vertical line of action. Transverse Metacenter: Roll/Heel Longitudinal Metacenter: Pitch/Trim The concept of the metacenter is only really valid for small angles of motion. Metacentric height is the distance measured from the metacenter to the center of GMgravity, . If the metacentric height is large, then the vessel is considered to be “stiff” in roll – indicating that there will be a large righting moment as a result of small roll angles. In contrast, if the metacentric height is small then the ship rolls slowly due to a smaller righting arm. docsity.com