MHMT 5 Momentum Heat Mass Transfer Inspection analysis

MHMT 5 Momentum Heat Mass Transfer Inspection analysis Flow resistance Inspection analysis of the Navier-Stokes equation. Dimensionless criteria Re, St, Fr. Friction factor at internal flows, Moody’s diagram. Drag coefficient at flow around objects. Karman vortex street. Taylor’s bubble. Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010

MHMT 5 Dimensionless NS equations Navier Stokes equations can be solved analyticaly (only few simple cases) or numericaly (using CFD software). However some information can be obtained without solving NS equation by using INSPECTION ANALYSIS Navier 1785 Stokes 1819 -1903 -1836 Navier Stokes equation Substitute actual values by characteristic (mean) values [N/m 3] transients convective acceleration pressure forces viscous forces gravity forces Number of variables describing the problem (7 – L, u, t, , , p, g) can be grouped to 5 terms, characterizing types of acting forces.

MHMT 5 Inspection analysis of NSeq. Ratios of these terms are dimensionless quantities, characteristic numbers determining relative influence of corresponding forces Reynolds number (convective inertial/viscous forces). This criterion is used for prediction of turbulence onset. Basic criterion for all phenomena with viscous forces Strouhal number (transient inertial forces / convective acceleration forces) Strouhal 1850 -1922 Criterion is used for prediction of eigenfrequencies of induced oscillations (vortex detachment in wakes). You can call it dimensionless frequency. Fourier number (viscous forces / transient inertial forces). Transient phenomena, see also penetration depth. You can call it dimensionless time. Fourier 1768 -1830 Froude number (convective acceleration forces / gravity forces) Criterion is used for prediction of free surface flows. Square root of (g. L) is velocity of surface waves in open channels. Froude 1810 -1879 Euler number (pressure forces / inertial forces). You can call it dimensionless pressure drop.

MHMT 5 Re-Reynolds number is probably the most important criterion affecting all transport phenomena. Flow resistance and drag coefficient, important for calculation of trajectories of droplets, sedimentation, settling velocities are just examples. Magritte

MHMT 5 c. D(Re) flow around objects In terms of dimensionless criteria it is possible to summarize results of experiments (real experiments or numerical simulations) into graphs or into simple engineering correlations. Reynolds number is used for example in correlations for drag coefficient, necessary pro prediction of hydrodynamic forces at flows around bodies. Newton’s law Projected surface Drag coefficient Dynamic pressure Osborn Reynolds 18421912 Drag coefficient reflects the action of viscous and pressure forces (friction and shape factor). D’Alembert’s paradox. Analytical solutions based upon Euler equation (see previous lecture) indicate, that the resulting pressure force (integrated along the whole surface of body) should be zero, and because fluid is inviscid, the overall drag force must be zero. Stokes derived the drag reduction for sphere (c. D=24/Re) taking into account viscosity, however this solution extended to very high Reynolds number predicts also zero resistance (because for large Re the Navier Stokes equations reduce to Euler equations). This discrepancy was resolved by Ludwig Prandl by introducing the concept of boundary layer, see next lecture.

MHMT 5 c. D(Re) plate Drag force on PLATE (length L, width 1) at parallel flow u Laminar flow regime Lcrit L Turbulent Re. L > 500000 Notice the difference between the mean and local values (mean value is twice the local value) wall shear stress profile this is the Blasius solution, which will be discussed in the next lecture (Blasius was Prandtl’s student) Transition of boundary layer to the turbulent flow regime at distance Lcrit (this distance decreases with increasing velocity) has several important consequencies, for example the DRAG CRISIS described in the next slide…

MHMT 5 c. D(Re) sphere Drag force on SPHERE (diameter D) Drag crisis at critical Recrit=3. 7 105. The sudden drop of resistance is caused by shifted separation point of the turbulent boundary layer. Figures calculated by CFD describe distribution of Reynolds stresses xx and indicate position of the separation point. D Oseen Stokes George Constantinescu, Kyle Squires: Numerical investigations of flow over a sphere in the subcritical and supercritical regimes. Phys. Fluids, Vol. 16, No. 5, May 2004

MHMT 5 c. D(Re) sphere creeping flow Stokes solved also the case of rotating sphere SPHERE A brief outline of the Stokes solution of the creeping flow regime when Re<1 u r ur U Navier Stokes equations should be written in the spherical coordinate system. In view of symmetry only the equations for radial and tangential momentum transport are necessary. Convective acceleration terms are neglected (Re<<1) therefore resulting equations are linear R u Stokes u potential flow Continuity equation in the spherical coordinate system completes the system of equations (3 equations for ur u p) Velocities can be approximated in a similar way like in the potential solution by and pressure can be eliminated from NS equations. Resulting ordinary differential equations for (r) and (r) can be solved analytically and together with boundary conditions (zero slip velocity at wall) give the velocity fields …compare with the solution for potential flow Pressure profile p( ) and the viscous stresses r on the sphere surface can be calculated by integration of NS equation p/ = (…). Resulting force is obtained by integration of pressure (Fp=2 RU) and viscous stress component (F =4 RU).

MHMT 5 c. D(Re) sphere and bubbles There is no unique and simple description of the sphere motion, because different forces act at laminar and turbulent flows. While at lower Re values the viscous resistance force prevails (see the Stokes solution when the form resistance is only ½ of friction resistance), at higher Re, in the inertial region, the boundary layer is separated from the sphere surface and a wake is formed, accompanied by the prevailing form drag (c. D=0. 44). The drag falls down (c. D=0. 18) when the boundary layer becomes turbulent (Recrit=3. 7 105) and the point of separation is shifted, thus reducing the region of wake. Previous analysis (and previous graph) is valid only for steady motion of a solid spherical particle. In the case of accelerating particles another resistance caused by inertia of surrounding fluid is to be considered (virtual mass of fluid Mf/2 is to be added to the mass of particle). Also the so called Basset forces, corresponding to acceleration of boundary layer should be respected. Quite another forces act on small bubbles or spheres filled by fluid (see also Taylor’s spherical caps which will be analyzed later). In this case the shear stresses on the surface are reduced and so the drag coefficient

MHMT 5 c. D(Re) Rotationally symmetric bodies Laminar flow regime u Momentum flux u 2 Turbulent Re. L > 500000 x Professor Fred Stern Fall 2010

MHMT 5 Re-friction factor (Moody’s diagram) Probably the most frequent problem for a hydraulic engineer is calculation of pressure drop in a pipe, given flowrate and dimensions, therefore given Reynolds number Re Please be aware of the relation between the Fanning friction factor f and the d. Arcy Weisbach friction coefficient f=4 f Stuart W. Churchill (turbulent flow, wall rougness e is respected) laminar Blasius (hydraulically smooth pipe and Re<105)

MHMT 5 St-Strouhal number describes frequencies of flow pulsation which is manifested for example by the „singing” wires in wind Magritte

MHMT 5 St-Strouhal and KARMAN vortex street Karman vortex street is a repeating pattern of swirling vortices caused by the unsteady separation of boundary layer on surface of bluff bodies. The regular pattern of detached vortices is typical for 2 D bodies like cylinder and not for a sphere (however even in this case vortex rings are formed). Von Karman vortex street behind a cylinder will only be observed above a limiting Re value of about 90. Dimensionless frequency of the vortices detachment is the Strouhal number. D St-Strouhal number 250 < Re < 2 × 105 (however, the Karman vortex street exists also at laminar flow regime)

MHMT 5 St-Strouhal and KARMAN vortex street Von Karman described the vortes street by two mutually shifted rows of counterrotated vortices (circulations described by circulation potentials, ). This kind of analysis is quite complicated. However, qualitative information about the frequency of vortices shedding can be obtained by inspection analysis of the vorticity transport equation This kind of correlations are used both in laminar and turbulent flow regime, see e. g. Aref

MHMT 5 Fr-Froude number describes velocity of gravitational waves on surface of fluid (velocity of shallow water waves, free surface in stirred vessels in the presence of gravitational and centrifugal forces) Magritte

Fr-Froude and TAYLOR bubble MHMT 5 Taylor bubble problem concerns calculation of rising velocity of a large volume of gas, such as those produced in submarine explosion. In peacetime are probably more important applications for heterogeneous flows, e. g. motion of large steam “slugs” in vertical pipes (slug regime at flow boiling). spherical cap wake U G. I. Taylor: published papers on underwater explosions R mention the fact, that the velocity U of rising bubble is independent of densities and depends only upon the radius R of spherical cap R. M. Davies and G. I. Taylor, “The mechanics of large bubbles rising through liquids in tubes”, Proc. of Roy. Soc. , London, 200, Ser. A, pp. 375 -390, 1950.

MHMT 5 Fr-Froude and TAYLOR bubble Proof: Solution of potential flow around a sphere gives velocity at surface U x R uw Distribution of pressure on the surface of sphere (see lecture 2) unlike solid particles (spheres) there is nonzero velocity at surface Pressure inside the sphere must be constant (p 0 there is only a gas) therefore variability of pressure along the bubble surface must be compensated by gravity R. M. Davies and G. I. Taylor, “The mechanics of large bubbles rising through liquids in tubes”, Proc. of Roy. Soc. , London, 200, Ser. A, pp. 375 -390, 1950. and this is the result presented in the previous slide

MHMT 5 EXAM Inspection analysis

MHMT 5 What is important (at least for exam) Definition and interpretation of dimensionless criteria dimensionless velocity dimensionless frequency dimensionless time gravity waves dimensionless pressure drop

MHMT 5 What is important (at least for exam) Drag forces Plate (Blasius) Sphere (Stokes) These expressions hold only in the creeping flow regime Cylinder (Lamb)