Cours(Gervais_2013)

Fluid Mechanics November 2013 Chapter Introduction and Fundamental Principles 1.1 Introduction One calls fluid a material medium which can be (largely) deformed from very weak efforts (or stress) Contrary to the solids, there is not minimal threshold of stresses to obtain a given deformation Problems encountered are varied It is often important to know how the fluid flows around solid bodies, and the knowledge of pressures and flow velocities in the vicinity of the walls is particularly useful One also often seeks to determine the stresses, the heat fluxes, the fluxes of chemical species, or integrated quantities such as hydrodynamic forces or moments 1.2 Concept of continuous medium One is interested in the movements of sets (material volumes) made of a very high number of molecules The characteristic dimensions of the problems considered are very large compared to the average distance between the molecules (mean free path) 1.2.1 Density Let us consider a continuous fluid medium inside a volume δV around a point P, δm being the mass of this elementary volume The density, or more precisely the volume mass (mass per unit of volume) is defined by: ρ= lim δV→δVlim δm δV (1.1) where δVlim is the limit volume, large compared to the mean free path At the macroscopic scale, δVlim defines the characteristic volume of a particle, which can be considered as a point at the macroscopic scale The matter is then considered as being formed of an endless number of particles, a particle being the smallest material volume on which it is still possible to apply the general laws and balance equations of classical mechanics (Mechanics of continuous media lemma) 1.2.2 Velocity in one point The velocity of the point P in a reference frame R is given by: −→ ! → − dOP V (P/R) = dt (1.2) R where P is here a particle, that is as previously said a very small elementary volume, but very large at the microscopic scale At the macroscopic scale, P is a point 1.2.3 Stress Let us consider a fluid volume V, a point P of the surface Σ of this volume, an − element of this surface (elementary surface) dσ around P of external normal → n The fluid external to V exerts forces (strengths, efforts) on dσ, which generate stresses The latter are defined by − → → −τ = lim dF = → − − τn + → τt dσ→0 dσ (1.3) − → where dF is the element of force (or effort) exerted on V by the external medium −τ into two parts: One can break up → − • a normal part → τn − • a tangential part → τt The normal part (the normal stress) represents the effect of the pressure whereas the tangential part (the tangential stress) corresponds to the friction 1.3 Different characteristic flows One gives here a short qualitative review of some types of flows 1.3.1 Incompressible and compressible flows Incompressible fluid: One considers a fluid incompressible if its mass per unit of volume (density) is constant or varies only a little with pressure or temperature (⇔ ρ = cst) Water flows are an example as they are generally considered as incompressible Compressible fluid: the opposite As a general example, gases (at high velocities) are considered as compressible fluids 1.3.2 Laminar and turbulent flows One says that a flow is laminar when the motion of the particles is regular and ordered The flow is turbulent when the displacement is irregular and when fluctuations are superimposed to the mean motion of the fluid A well known example is given by the natural convection flow of the smoke of a cigarette (see figure) In order to precise the conditions of turbulent flow achievement a few classical experiments can be shortly described Flow in a duct: Reynolds experiment (1883) Let us consider a duct connected to a tank containing a fluid under pressure, and a thin tube allowing the injection of coloured fluid into this duct (see figure) When the velocity of the flow is low, the streamlines of the fluid are stable, regular and ordered If one increases the velocity of the fluid, the motion of the particles becomes irregular, the particles being of course convected by a mean velocity The experience shows that the problems depends in fact of parameters: the density ρ, the velocity V, the diameter of the duct D and the (dynamic) viscosity of the fluid µ From these parameters it is possible to define the following dimensionless number, the Reynolds number: Re = ρVD µ (1.4) One can notice that in this situation the transition from the laminar to the turbulent regime takes place when Re is about 2000 to 2500: • If Re < 2000, the flow remains laminar, • From Re = 2000 to 2500, some turbulent bursts appear, • If Re > 3000, the flow is fully turbulent Flow around a cylinder It is still possible to define the Reynolds number, based here on the diameter of the cylinder • From Re about 10 to 40, one can notice two areas of re-circulation in the wake of the cylinder These two areas are stretched (and lie) when the Reynolds number increases, • From Re about 40, on can notice the development of an instability within the wake Beyond this value of the Reynolds number, one can observe alternated vortices in the wake, • At Re about 100, the vortices separate from the cylinder at regular intervals (Von Karman street) When the Reynolds number becomes very high the wake is fully turbulent 1.3.3 Steady and unsteady flows A flow is steady if all the variables that describe the motion are independent of time On the opposite, if one variable depends on time the flow is unsteady All the wave propagation phenomena are unsteady, as well as atmospheric flows All the turbulent flows are unsteady by definition However, one can consider that a turbulent flow is steady if the mean variables that describe the flow are independent of time 1.3.4 Unidimensional flows One says that a flow is unidimensional if the variables are constant on each section of the flow In other words, all the variables are dependent of a space coordinate only In practical applications, many flows can be considered unidimensional in a first approximation, leading to interesting simplifications especially when considering in duct flows Chapter Static of fluids Before to deal with the dynamic behaviour of flows, one starts to study the general properties of fluids at rest One says that a fluid is at rest in a Galilean reference frame if the velocity field is null at any point of the fluid and if all the mechanical and thermodynamical quantities that characterise the state of the fluid are independent of time 2.1 General equation of the static of fluids − Let us consider a fluid volume V bounded by a surface Σ of external normal → n The momentum equation for this fluid volume at rest is: → − Σ F ext = (2.1) The external forces exerted on this volume are of two kinds: the surface forces and the volume forces Surface forces In any point M of the surface Σ of the volume V of the fluid at rest, the −τ (surface force) reduces to its normal component only, so that one can stress → write: → −τ (M, → − − n ) = −p(M)→ n (2.2) Effectively, if there is no motion, there are no tangential forces on the surface of the considered fluid volume (friction is null), and the only forces that can exert on V are normal forces Let us notice here that p is nothing else that pressure We then obtain for the whole surface Σ of the volume V: ZZ − → − FS = − p(M)→ n dσ (2.3) Σ Volume forces In any point M of the volume V submitted to distant forces (volume forces) → − of volume density f , the strength exerted in M can be written → − → − f = ρφ (2.4) → − where φ is a mass density of force We then obtain for the whole volume V: ZZZ −→ → − FV = ρ φ dv (2.5) V Static equation The equation of static (2.1) yields: − → −→ FS + FV = which leads to: ZZ − − p(M)→ n dσ + (2.6) → − ρ φ dv = ZZZ (2.7) V Σ The generalised Green-Ostrogradsky relation allows to write: ZZZ ZZZ −−→ → − − grad p dv + ρ φ dv = V (2.8) V which can still be written as : ZZZ −−→ → − (− grad p + ρ φ )dv = (2.9) V If this equation is right for a volume V, it is right for any volume V and especially for an elementary volume dv (lemma of the mechanics of continuous media) The previous equation can then be reduced to its following local expression: −−→ → − − grad p + ρ φ = 2.2 2.2.1 (2.10) Incompressible fluid in the gravity field General equation → − In this (common) situation, the mass density of force φ reduces to the gravity → − g so that we have: → − → − − φ =→ g = −g k (2.11) → − where k is an ascendant vertical vector of unity norm If the fluid is considered as incompressible, its density is constant (ρ = cst) and it comes: −−→ −−→ ∂(gz) → → − − − − ρ φ = ρ→ g = −ρg → z = −ρ z = −ρ grad (gz) = − grad (ρgz) ∂z (2.12) With the equation (2.10), it comes: −−→ −−→ − grad p − grad (ρgz) = (2.13) so that the static equation for an incompressible fluid at rest is given by: p + ρgz = cst 2.2.2 (2.14) Consequences a - Isobar surfaces are in horizontal plans This leads to the principle of connected vessels: In connected vessels of different shapes containing a fluid at rest, free surfaces are in a same horizontal plan b - The quantity p is called the absolute static pressure Its dimension is: [p] =ML−1 T−2 (with M the mass, L the length and T the time) In the MKS International System (SI), p is expressed in Pascal, noted Pa We also find as unit of pressure: • The Bar : Bar = 105 Pa • The Atmosphere : Atm = 1.01325 105 Pa c - One can also define the driving pressure p∗ : p∗ = p + ρgz (2.15) The static equation (2.14) then leads to: p∗ = cst (2.16) for a fluid at rest d - We call effective pressure the quantity pe = p − patm involving in that case the atmospheric pressure to be taken as reference This involves possible negative values for pe e- Pressure measurement: Consider a tank containing water of density ρe , a point A of the free surface (at the atmospheric pressure) and an another point M at depth h The fluid is at rest and from the static equation (2.14) one can write: pA + ρe gzA = pM + ρe gzM (2.17) peM = pM − patm = ρe gh (2.18) that is The knowledge (or the measurement) of h allows to obtain the effective pressure at point M 2.2.3 Incompressible fluid in relative equilibrium − For a fluid in relative equilibrium, the relative acceleration is null (→ γ rel = 0) → − but the driving acceleration is not ( γ ent 6= 0) The Fundamental Principle of the Dynamics then leads to (per unit of volume, that is for the local form of the static equation): −−→ − − − grad p + ρg → z + ρ→ γ ent = (2.19) → − with z the vertical ascendant vector Example : Water in a tank in linear uniformly accelerated motion in the direc− tion → x − The absolute acceleration of the tank → γ abs reduces to its driving acceleration → − γ ent Thus we have: −−→ − − grad p + ρg → z + ργ→ x =0 (2.20) (with γ = γent ) Then we obtain:  ∂p   + ργ =   ∂x    ∂p =0  ∂y       ∂p + ρg = ∂x As at x = and z = zo the pressure p is equal to the atmospheric pressure pa , one shows that: p(x, z) = pa − ργx + ρg(zo − z) (2.21) and that the equation z = z(x) of the free surface is given by, as it is at the atmospheric pressure pa too: γ z(x) = − x + zo g 2.3 (2.22) Static of a compressible fluid in the gravity field In this situation, the density ρ is no more a constant and we have to take into account its variations (with z) The static equation is still given by: −−→ − grad p + ρg → z =0 with ρ 6= cst Thus: (2.23)  ∂p   =0   ∂x    ∂p =0  ∂y       ∂p = −ρg ∂z which then reduces to: dp = −ρg dz (2.24) with ρ 6= cst Case of the perfect gas The state relation for a perfect gas is given by: p = rT ρ (2.25) with for air r = 287 sm2 K (perfect gas constant) The equation of static then becomes: dp p = −ρg = − g dz rT (2.26) g dp = − dz p rT (2.27) which leads to: In order to close the system (appearence of the temperature T in the expression of dp p ), it is necessary to introduce another equation It is an equation that traduces the nature of the thermodynamic transformations: Constant temperature T = To : gz p(z) = p(z=0) e− rTo (2.28) Isentropic transformations (adiabatic + reversible): p = cst ργ 2.4 2.4.1 Actions of pressure on an immersed body at rest - Archimède’s theorem Pressure forces − Consider a body of volume V bounded by a surface Σ of external normal → n immersed into a fluid, M being a current point of its surface As the body is at