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Lecture 4 Pressure distribution in a fluid Pressure and pressure gradient Mecânica de Fluidos Ambiental 2015/2016 Chapter 2 • • • • Concept of pressure and pressure gradient Hydrostatic pressure distribution Hydrostatic forces Buoyance and stability Mecânica de Fluidos Ambiental 2015/2016 Forces in fluids • Surface or volumetric (or mass) • Surface forces can be normal (pressure) or tangential (friction) • Friction Forces are always parallel to velocity. This is why they are often called “tangential forces”. Their equation can be complex if velocity is not parallel to any reference axis. Mecânica de Fluidos Ambiental 2015/2016 Pressure • Many fluid problems do not involve motion. They concern the pressure distribution in a static fluid and its effect on solid surfaces and on floating and submerged bodies. • Fluids at rest cannot support shear stress. • Pressure is used to indicate the normal force per unit of area at a given point acting on a given plane within the fluid mass of interest. • How the pressure at a point varies with the orientation of the plane passing through the point? Mecânica de Fluidos Ambiental 2015/2016 Pressure at a point • Small wedge (“pequena cunha”) of fluid at rest of size ïx by ïz by ïs and depth b into the paper. • There is no shear by definition, but we postulate that the pressures px, pz, and pn may be different on each face • The weight of the element also may be important. • The element is assumed small, so the pressure is constant on each face. • Summation of forces must equal zero (no acceleration) in both the x and z directions. Mecânica de Fluidos Ambiental 2015/2016 Pressure at a point ðđðĨ = 0 = ððĨ ðâð§ − ðð ðâð sin ð 1 2 ðđð§ = 0 = ðð§ ðâðĨ − ðð ðâð cos ð − ðððâðĨâð§ By the geometry of the wedge: âð sin ð = âð§ âð cos ð = âðĨ ððĨ = ðð 1 ðð§ = ðð − 2 ððâð§ These relations illustrate two important principles of the hydrostatic, or shear-free, condition: (1) there is no pressure change in the horizontal direction (2) there is a vertical change in pressure proportional to the density, gravity, and depth change In the limit as the fluid wedge shrinks to a “point”, âð§ ððĨ = ððĶ = ðð§ = ð 0: Since ïą is arbitrary, we conclude that the pressure at point (p) in a static fluid is a point property, independent of the plane orientation. Mecânica de Fluidos Ambiental 2015/2016 Pressure force in fluid element • Pressure (or any other stress, for that matter) causes a net force on a fluid element when it varies spatially. • Consider the pressure acting on the two x faces in Fig2.2 and let the pressure vary arbitrarily: ð = ð ðĨ, ðĶ, ð§, ðĄ • The net force in x direction on the element is: ðð ðð ððđðĨ = ðððĶðð§ − ð + ððĨ ððĶðð§ = − ððĨððĶðð§ ððĨ ððĨ • The total net-force vector on the element due to pressure is: ððđðððð ð = −ð ðð ðð ðð −ð −ðĪ ððĨððĶðð§ ððĨ ððĶ ðð§ Mecânica de Fluidos Ambiental 2015/2016 Equilibrium of fluid element • Forces acting on the fluid element: surface forces due to the pressure, body force equal to weight of the element and friction ððĶðĨ ððĶðĨ ðĶ ≈ ð ðð ððĶ ðĶ ðð ððïŪ = lim ð ððĶ 0 ððĶ ðð ððĶ ðĶ+ððĶ Weight=ïēg(dxdydz) −ð ðĶ+ððĶ ðð ððĶ By volume unit and considering ï constante: Mecânica de Fluidos Ambiental 2015/2016 ðĶ+ððĶ ≈ ð = ðĶ ð ðð ð ððĨððĶðð§ ððĶ ððĶ ðïŪð = ð ð2 ð ððĨ 2 + ð2 ð ððĶ2 + ð2 ð ðð§ 2 = ððŧ 2 ð Equilibrium of fluid element • ððððð ð = −ðŧð (net pressure force per unit volume) • ðððððĢ = ðð (net gravity force per unit volume) • ðïŪ = ððŧ 2 ð(net viscous force per unit volume) • By Newton’s law, the sum of these per-unit-volume forces equals the mass per unit volume (density) times the acceleration a of the fluid element: ð = ððððð ð + ðððððĢ + ðïŪ = −ðŧð + ðð + ððŧ 2 ð = ðð Mecânica de Fluidos Ambiental 2015/2016 General equation ðð = −ðŧð + ðð + ððŧ 2 ð ðð ð ððĄ = −ðŧð + ðð + ððŧ 2 ð • If the fluid is at rest or at constant velocity, a=0 and ðŧ 2 ð = 0 the equation for the pressure distribution reduces to ðŧð = ðð This is a hydrostatic distribution and is correct for all fluids at rest, regardless of their viscosity, because the viscous term vanishes identically. Mecânica de Fluidos Ambiental 2015/2016 Hydrostatic pressure • In our customary coordinate system z is “up”. Thus the local-gravity vector for small-scale problems is: ð = −ðð where g is the magnitude of local gravity, for example, 9.807 m/s2. For these coordinates, equation ðŧð = ðð has the components ðð ðð ðð =0 =0 = −ðð = −ðū ððĨ ððĶ ðð§ • These equations show that pressure does not depend on x and y. Thus, as we move point to point in horizontal plane, pressure do not change. Since p depends only on z, ðð ðð§ can be replaced by the total derivative ðð ðð§ Mecânica de Fluidos Ambiental 2015/2016 = −ðð = −ðū Hydrostatic pressure ðð ðð§ = −ðū or ð2 −ð1 = 2 −ðū 1 ðð§ • These equation is the solution to the hydrostatic problem. The integration requires an assumption about the density and gravity distribution. Gases and liquids are usually treated differently. • Pressure in a continuously distributed uniform static fluid varies only with vertical distance and is independent of the shape of the container. • The pressure is the same at all points on a given horizontal plane in the fluid. • The pressure increases with depth in the fluid. Mecânica de Fluidos Ambiental 2015/2016 Efect of variable gravity • For a spherical planet of uniform density, the acceleration of gravity varies inversely as the square of the radius from its center ð0 2 ð = ð0 ð where r0 is the planet radius and g0 is the surface value of g. For earth, r0 ïŧ 3960 statute mi ïŧ 6400 km. In typical engineering problems the deviation from r0 extends from the deepest ocean, about 11 km, to the atmospheric height of supersonic transport operation, about 20 km. This gives a maximum variation in g of (6400/6420)2, or 0.6 percent. We therefore neglect the variation of g in most problems. Mecânica de Fluidos Ambiental 2015/2016 Hydrostatic pressure in liquids • Liquids are so nearly incompressible that we can neglect their density variation in hydrostatics. Thus we assume constant density in liquid hydrostatic calculations, for which hydrostatic conditions integrates to ð2 − ð1 = −ðū ð§2 − ð§1 or ð§1 − ð§2 = ð2 ðū ð1 − ðū We use the first form in most problems. The quantity ï§ is called the specific weight (“peso específico”) of the fluid, with dimensions of weight per unit volume; The quantity ð ðū is a length called the pressure head (“carga de pressão”) of the fluid. Mecânica de Fluidos Ambiental 2015/2016 Hidrostatic pressure distribution • For lakes and oceans, the coordinate system is usually chosen as in Figure, with z=0 at the free surface, where p equals the surface atmospheric pressure pa. ð1 , ð§1 = ðð , 0 Mecânica de Fluidos Ambiental 2015/2016 ð = ðð − ðūð§ Hydrostatic pressure in gases • Gases are compressible, with density nearly proportional to pressure. Thus density must be considered as a variable in the hydrostatic pressure equation. It is sufficiently accurate to introduce the perfect-gas law ð = ðð ð in the hydrostatic equation: ðð ð = −ðð = − ð ðð§ ð ð Separate the variables and integrate between points 1 and 2: 2 ðð ð2 ð 2 ðð§ = ln = − ð1 ð 1 ð 1 ð The integral over z requires an assumption about the temperature variation T(z). One common approximation is the isothermal atmosphere, where T= T0: ð ð§2 − ð§1 ð2 = ð1 ððĨð − ð ð0 Mecânica de Fluidos Ambiental 2015/2016 Hydrostatic forces on plane surfaces • Objective: compute the hydrostatic force over flat and curve surfaces and the application point (center of pressure). • When a surface is submerged in a fluid, forces develop on the surface due to the fluid. The determination of these forces is important in the design of storage tanks, ships, dams, and other hydraulic structures. Mecânica de Fluidos Ambiental 2015/2016 Hydrostatic forces on plane surfaces • Plane panel of arbitrary shape completely submerged in a liquid • Panel plane makes an arbitrary angle ïą with the horizontal free surface, so that the depth varies over the panel surface. • If h is the depth to any element area dA of the plate, from hydrostatic pressure equation in liquids the pressure there is p=pa+ï§h. Mecânica de Fluidos Ambiental 2015/2016 Hydrostatic forces on plane surfaces • To derive formulas involving the plate shape, establish an xy coordinate system in the plane of the plate with the origin at its centroid, plus a dummy coordinate ïļ down from the surface in the plane of the plate. A is the area of the plane surface. • The total hydrostatic force on one side of the plate is given by From the figure â = sin ð and, by definition, the centroidal slant distance from the surface to the plate is Mecânica de Fluidos Ambiental 2015/2016 • Therefore, since ïą is constant along the plate, becomes • Finally, unravel this by noticing that , the depth straight down from the surface to the plate centroid. Thus • The force on one side of any plane submerged surface in a uniform fluid equals the pressure at the plate centroid times the plate area, independent of the shape of the plate or the angle ïą at which it is slanted. Mecânica de Fluidos Ambiental 2015/2016 Center of pressure • However, to balance the bending-moment portion of the stress, the resultant force F acts not through the centroid but below it toward the high-pressure side. Its line of action passes through the center of pressure CP of the plate (see figure). • To find the coordinates (xCP, yCP), we sum moments of the elemental force p dA about the centroid and equate to the moment of the resultant F. To compute yCP, we equate Vanishes by def. of centroidal axes Ixx is the area moment of inertia of the plate area about its centroidal x axis Mecânica de Fluidos Ambiental 2015/2016 Center of pressure • The negative sign shows that yCP is below the centroid at a deeper level (below the gravity center) and, unlike F, depends on angle ïą. • The determination of xCP is exactly similar: Ixythe product of inertia of the plate, again computed in the plane of the plate • If Ixy = 0, usually implying symmetry, xCP= 0 and the center of pressure lies directly below the centroid on the y axis. Mecânica de Fluidos Ambiental 2015/2016 • In most cases the ambient pressure pa is neglected because it acts on both sides of the plate; for example, the other side of the plate is inside a ship or on the dry side of a gate or dam. In this case pCG=ï§hCG, and the center of pressure becomes independent of specific weight: Mecânica de Fluidos Ambiental 2015/2016 Resume Mecânica de Fluidos Ambiental 2015/2016