Fundamentals of Fluid Dynamics

Transcription

1 Chapter Fndamentals of Flid Dynamics - Flid Dynamics of Ocean and Atmosphere Laminar flow : orderly flow Inviscid : Lacking viscos forces Internal Stress : Forces per nit area on the flid at any point de to the adacent flid Vorticity : A measre of the anglar velocity or spin of the flid at a point Geophysical flow and classical flid dynamics have two basic differences in emphasis: () Geophysical flow is moving in a rotating frame of reference, e.g., earth. () Trblence is an essential part of geophysical flid dynamics. - Newton s Law in a Rotating Frame of Reference Newton s law in classical mechanics: where F = ma F is the net force m is the mass a is the acceleration The significant forces in geophysical flid are F = Σ F = F p F g F τ = m a (.) where F p is the force de to pressre differences F g is the force of gravity F τ is the internal friction force For most atmospheric and oceanic flow problems, the preferred coordinate is the Cartesian coordinate fixed to the srface of the earth. It is an (x, y, z) Cartesian coordinate system with the x-y plane tangent to the earth s srface and the z direction denoting the height above this plane as shown in Fig... -

2 Figre. Description of P in the rotating coordinate system r(x, y, z) and the fixed coordinate system r a. The distance r is greatly exaggerated for clarity. In order to se the Newton s law to describe the geophysical flow in a rotating frame of reference, the concept of virtal force will be sed. In other words, to write Newton s law in the non-inertial earth-based coordinate system, we will have a new force balance, d F = F p F g F τ F ν = m a = m (.) dt where and F ν is the virtal force (Coriolis force) F g = F g F Ce F Ce is the centrifgal force Example.: Consider the pendlm shown in Fig... (a) Based on yor intitive feeling for acceleration, draw in the acceleration vectors at each of the positions of the ball on the end of a string in Fig... (b) Then derive the direction of the acceleration by considering the incremental change in velocity at each point. (c) Finally, consider Eq. (.) as the definition of acceleration. Figre. A pendlm consisting of a ball sspended on a (rigid, weightless) string. Soltion: (b) Intitive ideas of acceleration generally are associated with velocity, althogh the distinction between velocity and velocity increment sometimes gets lost. From this standpoint, the acceleration at the high points of Fig.. is sometimes thoght to be zero. However, when an incremental velocity change -

3 is obtained by considering the velocity vector at the next instant, the correct acceleration vectors qickly reslt (Fig..3a). (c) When the acceleration vector is seen as simply a vector in the opposite direction of the net force, the vector sm of the gravity force pls the force in the string yields the acceleration direction (Fig..3b). In one case the acceleration is determined as the incremental change in velocity; in the other it is the force per nit mass. There are evidently several ways of looking at the problem, and some prodce easier soltions than others. Formlation in terms of fndamental definitions is often the safest and srest rote to a soltion. We will develop the flid dynamic version of eqation. in Chapter 6. There, the balance will often be written in terms of force per nit mass, so that the eqation is a balance of accelerations. Figre.3 (a) The acceleration of the pendlm ball. (b) The forces and acceleration on the pendlm ball. Example.: Consider a ball that was proected in the air somehow and is now flying throgh the air. (a) Show the important forces on it. (b) If it is whirled on a string at a constant speed sch that F Ce >> F g, show the forces on the ball. (c) If the coordinate system is now rotated so that the ball appears to stand still, write the balance for Newton s law. Soltion (a) The main force on the ball will be the force of gravity and a friction force de to air resistance. We know that the first force will assre the ball s retrn to earth. The second force slows obects p. We can assme that the latter acts in the opposite direction to the velocity (Fig..4) -3

4 Figre.4 Forces (de to gravity & air friction) on a ball in free flight with velocity U. The forces do not balance, and according to Newton s relation, Σ F = m a, there will be an acceleration in the same direction as Σ F. Position vector v P = ( r cosθ, r sinθ ) v ĵ v dp dθ V = = ( r sinθ, r cosθ ) dt dt v v v dθ = ω P, ω kˆ kˆ dt. î v v v v dv dω v v dp a = = P ω Figre.5 Forces (centrifgal, gravitational, and in the dt dt dt string)on a ball being swng on a string at velocity θ ( a. θ eˆ θ ) ( aneˆ n ) (b) When there is rotation we expect a new force to be v v v dθ present, called the centrifgal force. For a circle ω ω P = reˆ dt motion in the x-y plane at constant speed θ and radis r (length of the string), there is a normal acceleration, a n = r( dθ / dt). The rotation speed is θ = ds dθ = r. Hence, an = dt dt θ F Ce r m The force balance and the ball is in eqilibrim, as shown in Fig..5. We note that if the string were ct, the ball wold accelerate otward de to centrifgal (otward) force. ( ) n -4

5 Figre.6 forces on a ball swng on a string assming the gravitational force is relatively small. The coordinate system is rotating so the ball appears fixed. (c) When the coordinates is rotated abot the z-axis in Fig..6, to a person standing in the coordinate system the ball wold appear stationary. Example.3: Consider a tank that is half fll of liqid (Fig..7). If the tank is sitting on a rotating platform, discss and sketch the forces on a small box of flid extracted from the center of the liqid. Discss the forces and flow after the rotation is stopped. Figre.7 An imaginary small box of flid in a container of flid rotating in a large frame of reference. Soltion: A variation of pressre in the flid along the axis of the box exists becase there is more weight of water at the deeper sections. We know that the flid is piled p at one end de to the rotation. We can assme the air pressre on the top srface is constant since the difference in the weight of air above the ends of the box wold be insignificant. The pressre gradient prodces a force on the small box of flid, F p, which is acting on the box in the direction of lower pressre, toward the center of rotation. The box is rotating, therefore there is a centrifgal force directed away from the axis of rotation. In a rotating frame of reference we mst inclde a virtal centrifgal force F Ce, which acts to move the liqid otward. These two forces, F p and F Ce, constitte the primary horizontal force balance on the flid. Thogh gravitational force plls the liqid downward, this is balanced by an eqal and opposite force exerted by the floor. Friction will act on the box of flid to retard the motion, bt only when -5

6 F p = F ce. Ths, in the eqilibrim state, the two horizontal forces balance, while the vertical gravitional force is balanced by the floor spport, F floor (Fig..8). Figre.8 A close-p of the imaginary box of flid in a rotating flid. If the rotation is stopped, there will now be a velocity difference between the rotating flid next to the floor srface and the static floor. A friction force that is proportional to the velocity difference will propagate into the flid, slowing the flid down. As the flid slows, the centrifgal force (½U ) decreases, and the nbalanced pressre gradient force prodces acceleration toward the low pressre. The final state is reached when the flow has reached niform depth, U and F p are zero, and there is static eqilibrim. This example illstrates the different eqilibrim states for a rotating and nonrotating system. The pressre gradient force is reqired to balance the centrifgal force when the system is rotating. The expression of Newton s law is different nder different circmstances. These inclde 4 cases: () Horizontal components of velocity are mch greater than vertical, so that r Fg 0 and F p F Ce F τ = m d/ dt () The rotating frame of reference has negligible effect; conseqently, r FCe 0 and F p F τ F g = m d/ dt (3) There are no viscos effects --- Fτ 0 r F p F Ce F g = m d/ dt -6

7 (4) The acceleration is negligible (steady-state) and Σ F = 0, F p F Ce F g = 0-3 The Laminar Flow Regime and Potential Flow.3. Laminar Flow When the flow is orderly and predictable, it is called laminar flow. In laminar flow, the individal flid particles move along their traectories independent of the particles in the adacent layers. There are several types of laminar flow: niform flow => is parallel and constant everywhere shear flow => changes with one or more coordinates crving flow => has a constant or periodic change.3. Inviscid Flow An inviscid flid is one where the internal friction force is negligible compared to the other forces. Generally all flid have some internal friction that provides a resistance to motion. Flows with faster motion and small internal stress behave more inviscidly. The viscos behavior of some common flids is shown in Fig..9. Figre.9 Chart showing vales of internal resistance to flow of some specific flids. The internal stress of flid can be characterized by a qantity called viscosity. When the viscosity is small. The flow is called inviscid and the Newton s law for inviscid flow can be written as d F p F Ce F g = m (.3) dt Example.4: Horizontal geophysical flows above the srface (or below in the ocean) are often well described by a simple balance between the horizontal pressre gradient force and the inertial force arising from the rotating frame of -7

8 reference-the Coriolis force, F p F c = 0 (.4) These simple flows are called geostrophic. Discss the approximations made in Eq. (.) to obtain Eq. (.4). Soltion: Starting with a balance inclding all of the forces, apply the inviscid approximation, F p F g F τ F c = m a (Inviscid) 0 Now consider the flow as (horizontal-two-dimensional in a plane normal to the gravitational force), F p F g F c = m a (horizontal flow) 0 Finally, when the flow is steady, F p F c = m F c (steady-state flow) 0 This leaves the geostrophic force balance, F p F c = 0 Figre.0 The dye pattern of the flow past a circle. Sch a pattern reqires a specific flow environment (in this case a creeping flow of a high-viscosity flid in a narrow gap to approximate two-dimensional flow). (Photograph by D. H. Peregrine; from An Albm of Flid Motion, assembled by M. Van Dyke and pblished in 98 by Parabolic Press, Stanford, califormia.) -8

9 Figre. Symmetric plane flow past an airfoil. Streamlines are shown by colored flid introdced pstream in a water flow tnnel. [photograph cortesy of ONERA from H. Werk (974). Le Tnnel Hydrodynamiqe a Service de la Recherche A rospatiale, Pbl. No. 56, ONERA, France.].3.3 The Potential Fnction If a scalar fnction φ (x, y, z) satisfies this relationship, V = φ where V is the velocity field of the flid then φ is called the velocity s potential fnction, and the flow is called potential flow. The potential flows inclde 4 cases: () Parallel (Uniform) Flow () Linear Flow (3) Vortex Flow (4) Waves-Periodic motion -9

10 .4 Waves, Vortices, and Instabilities Some laboratory example of waves are shown in Figs.. and.3. Figre. A Karman vortex street behind a circlar cylinder. Streaklines are shown by electrolytic precipitation in water. (Photograph by Sadatoshi Taneda; from An albm of Flid Motion, assembled by M. Van Dyke and pblished in 98 by Parabolic Press, Stanford, California.) Figre.3 Boyancy-driven convection rolls. These are side views of convective instability patterns in silicone oil. At the top is the classical Rayleigh-B nard flow pattern for niform heating leading to rolls parallel to the shorter side. In the middle, the temperatre difference and hence the amplitde of motion increase from right to -0

11 left. At the bottom, the box is rotating abot a verical axis. [Photograph from H. Oertel Jr. and K. R. Kirchartz (979). Recent Developments in Theoretical and Experimental Flid Mechanics M. M ller, K. G. roesner, and B. Schmidt, eds.], pp Springer-Verlag, Berlin.] Atmospheric flows of wave examples are shown in Figs Figre.4 Satellite photograph (NOAA Nimbs 7) showing atmospheric flow with organized parallel streets of cmls clods sitting atop the planetary bondary layer. The flow is from over the oceanic pack ice (top) to over the sea, with clod street separation abot -3 km near the ice, 5-6 km at 00 km downstream. -

12 Figre.5 An example of von Karman vortices shown in the clod patterns downstream of Gadlpe Island off the coast of Baa Califormia. This skylab photograph shows a clodless area over the island, a cyclonic and anticylonic vorties immediately downstream, followed by two cyclonic vortexs. (photograph cortesy of O. M. Griffin, NRL.) Figre.6 A laminated layer of clods in the atmophere. -

13 Figre.7 A Voyager photo of Jpiter showing the Great red Spot and the trblent region srronding it. The smallest details seen are abot 00 km across. Figre.8 Laboratory flow of alminm flakes sspended in water past an inclined flat plate. The plate is several centimeters long and the Reynolds nmber is (Photograph by B. Cantwell, reprodced with permission from the Annal Review of Flid Mechanics 3; copyright 98 by Annal Reviews Inc.) -3

14 Figre.9 The tanker Argo Merchant agrond on Natcket shoals in 976. the ship is inclined abot 45 o to the mean crrent, and the leaking oil shows a wake pattern remarkably similar to Fig..8. Re ~ 0 7. (A NASA photograph cortesy of O. M. Griffin, Naval Research Laboratory.) Gravity waves or internal waves freqently occr in the atmosphere and ocean. A more descriptive term shold be sed if the case of the waves can be determined: boyancy waves for those cased by convective instabilities, topographic waves for those forced by flow over variable terrain, or dynamical instability waves..5 Trblence and Transition Trblence is an example of flid flow where the path of an individal parcel of flid is random and npredictable. Example of trblent fields are shown in Fig

15 Figre.0 Trblence being generated by a grid. Smoke shows laminar streamlines passing throgh a grid (-inch mesh size) and becoming trblent downstream. (Photograph by Thomas Corke and Hassan Nagib; from An Albm of Flid Motion, assembled by M. Van Dyke and pblished in 98 by Parabolic press, Stanford, California.) Trblence is a very important factor in the flow near any bondary where the air flow comes to rest. Sorce of trblence can also be fond at the edge of a rapidly growing clod, and at bondaries between adacent, different laminar flow regimes in clear air. There is also trblence on larger scales inclding the synoptic (~ 000km), where the random, large-scale eddies accomplish a net poleward heat transport. Figre. Homogeneos trblence behing a grid (0.-inch mesh size). At abot the middle of the photograph the merging nstable waves have formed an approximation of ideal isotropic trblence. (Photograph by Thomas Corke and Hassan Nagib; from An Albm of Flid Motion, assembled by M. Van Dyke and pblished in 98 by Parabolic Press, Stanford, California.) Both wave and vortex flow regimes can be waystations between laminar and trblent flows. Observations show that the realization of a particlar flow soltion, which may be pre laminar, wavy laminar, or trblent, depends on three characteristics of the flow. They are () the flow velocity, () the characteristic length scale, and (3) a measre of the internal ability of the flid to commnicate between layers via the net gravitational forces. -5

16 Example.5: It is of interest to see how varios sorces have approached the definition of trblence. Here are some definitions of and comments abot trblence. Webster s defines trblence as fll of commotion or wild disorder; violently agitated; marked by wildly irreglar motion. Trblent flow is defined as the random motion of layers of a flid, casing high resistance to movement throgh the flid. A mathematics book may give the following definition: Trblence a field of random or chaotic vorticity. Schlichting (960) wrote: It is not very likely that science will ever achieve a complete nderstanding of the mechanism of trblence becase of its extremely complicated natre.the most essential featre of a trblent flow is the fact that the pressre and velocity are not constant in time, bt exhibit very irreglar, high-freqency flctations. The velocity can only be considered constant on the average and over a longer period of time. Von Karman made the statement: To my mind, there are two great nexplained mysteries in or nderstanding of the niverse, one is the natre of a nified generalized theory to explain both gravitation and electromagnetism. The other is an nderstanding of the natre of trblence. After I die, I expect God to clarify general field theory for me. I have no sch hope for trblence. Saffman (98) wrote: Dring the past 45 years, mch effort has been spent trying to determine the statistical distribtion and in particlar the spectra of the vorticity distribtion in trblence. However, the most exciting recent development is the growing belief, sggested by modern experimental investigations, that the vorticity flctations are not qite so random or disorganized or incoherent as was commonly thoght. The vorticity is perhaps collected into coherent strctres or organized eddies, and it is now proposed that trblence shold be modeled or described as the creation, evoltion, interaction and decay of these strctres. Trblence is the thoght of as the random sperposition of organized, laminar, deterministic vortices, whose life history and relationships constittes the trblent flow. -6

17 .6 Bondary Layers The bondary layer is a region with at least one dimension that is very small compared to that of the average flow field. Generally, the bondary layer is fond in the regions adacent to a solid body. In this region, the flid experiences the layer-to-layer interaction that mst ltimately bring it to a halt at a srface. Ekman s eqations for the planetary bondary layer (PBL) assme steady-state, horizontal flow, so that F p F τ F Ce = 0 (.5) Ekman s bondary conditions reqire that there is a layer next to the srface where the flid velocity goes to zero. Ekmans soltion indicated that the effects of the srface decay exponentially with height; that is, the layer is thin. In general, the PBL has two fairly distinct regions. One is a strong shear region near the actal bondary (the srface of the earth or ocean), called the srface layer. In this region, the flow is in planes parallel to the srface, and the eddies are small and are expected to grow in size in proportion to distance form the solid srface. The second region inside PBL is a mch thicker layer, and the effect of earth s rotation becomes important. It is called Ekman layer, or the mixed layer. In this region, the eddies can be as large as the thickness of the layer. Example.6: Consider the flow in the PBL, from the top, z = H, where geostrophic flow prevails, to the bottom srface, where U 0. The Coriolis force is proportional to the wind speed, F C = fu, where f is a constant. Sketch the force balance on particles at varios heights in the PBL. The flow is horizontal and the pressre gradient force F, is constant. See Fig..3. p Figre.3 Velocity vectors at varios levels in the PBL. -7

18 Soltion: At geostrophic height (above the PBL) we can plot the force balance and the velocity in the x-y plane in Fig..4. Figre.4 (a) The geostrophic force balance between pressre gradient and Coriolos forces. U g is the geostrophic wind. In the pper PBL, the inflence of the srface is felt, slowing the wind slightly. The friction force F τ is small and in the opposite direction of the velocity. To have the vector sm of F C and F τ balance, the F C mst trn slightly conterclockwise. Since the Coriolis force mst be perpendiclar to U, 90 o to the right (northern hemisphere), it too mst trn: Figre.4 (b) Forces at the mid-level of a PBL. These now inclde the force of friction. In the lower PBL, F τ is increasing as the srface is approached. F C is decreasing as U is getting small. Since the angle of trning of U between U g and the srface is abot 0 o to 30 o, F is no longer opposite the U direction: τ Figre.4 (c) Forces near the srface of a PBL. The Coriolis force is qite small and the friction force mst nearly oppose the pressre gradient force. Note that the friction force is the gradient of the stress force, which may be nearly aligned with U. These are discssed in detail in chapter. -8

19 Finally, friction brings the flow to a halt [U(0) 0]. Since F τ is now balancing F p, it is nearly perpendiclar to U when last seen. However, the stress on the top and bottom sides of a slab of air are always approximately in the U direction. The stress force F τ is the gradient of this stress, st as the pressre gradient force F p, is the gradient of the pressre. The net reslt is that the viscos stress force acts to trn the flow throghot the PBL with height in the direction of F p, or toward low pressre. We will examine these forces on a parcel of flid in Chapter 6. Figre.5 Sketch of varios scale heights throgh the planetary bondary layer. Each higher scale presents a new regime and a new balance of forces in the governing eqations. Some characteristic scale parameters are shown. They emerge from the mathematical soltions for the layers. (Cortesy of Adam Hilger Ltd.) -9

20 Figre.6 Sketch of the planetary bondary layer winds. The winds trn and increase with height. Their proection on the srface plane is a hodograph. Typical heights are shown in meters. (Cortesy of Adam Hilger Ltd.) Figre.7 Sketch of dye marking transition from laminar to trblent flow in a Reynolds type experiment. -0

21 .8 The Flid Parcel We will consider flid properties pressre, temp, density associated with the parcel of flid. The parcel mst satisfy the following conditions:. Large enogh to contain sfficient molecles for a well-defined average of the properties.. Small enogh that the properties are niform across the parcel. 3. Uniqely identifiable for short periods of time. The concept of a flid parcel can be sed to define pathlines (traectories), streaklines, streamlines, which can be described as follows:. Identify a parcel with a spot of dye. The traectory of this parcel is a pathline.. Continosly inect dye at a point. The dye will mark a series of parcels that have occpied that point. This like is a streakline. 3. Streamlines are defined as being tangent to the direction of he flow at a given time. Figre.8 The flow lines with snapshots at times t 0, t, t, and t 3 of the flow from a leaking tank. A streamline, streakline, and pathline are shown. The behavior of a flid parcel at a point can be described sing the control volme approach, as shown in Fig..9. Figre.9 The parcel in the neighborhood of point P in a flow of variable pressre and velocity. The parcel is assmed to be a cbe at point P. -

22 .9 Continm and Averaging.9. Scale of the Domain In the description of the flid state, new properties defined as averages over a specified volme δ V are sed instead of individal moleclar mass and kinetic energy. Note that there is sally a vale of δ V where enogh molecles ae contained to make a meaningfl average, yet δ V can still be considered infinitesimal compared to the field dimensions..9. Continm Figre.30 The average density (mass pre nit volme) as the volme increases. Variation de to moleclar spacing for small volmes, de to environmental changes for large volmes. The continm hypothesis: A flid continm exists in the range of scales wherein variations in the macroscopic flid characteristics are small with respect to the mean variations, yet are not inflenced by the microscopic variations. The macroscopic flid characteristics are sally density, pressre, temperatre, and velocity. The microscopic motions are de to moleclar motion. The parcel is an abstract physical entity. We assme that all physical qantities are spread ot niformly over the volme of this continm parcel, and any parcel can be a representative of a series of parcels with the same properties. Note that the hypothesis of continm fails to exist when the flid dimension is close to the mean free path, the mean distance a molecle travels before hitting another molecle. The eddy-continm hypothesis: The continm is determined with respect to the small-scale trblent eddies rather than the molecles. Example.8: In applying the eddy viscosity assmption to the bondary layer regions, -

23 discss the allowable size of the eddies in order to have a well-defined vertical shear for both (a) the srface layer and (b) the entire PBL. For the srface layer, consider a depth of h ~ 0 4 cm, whereas the PBL has a scale of H ~ cm. Soltion: (a) To define the mean du/dz within the srface layer, δz (the parcel dimension) might be assmed to be small enogh at 0 3 cm, an order of magnitde smaller than the layer depth. However, this means that if we need 000 trblent eddies to determine a mean, then they mst be no larger than 0 cm, so that 000 will occpy a box with 0 3 -cm dimensions. If the eddies are larger, then a mean might still be defined by measring at a point for a sfficient interval of time. If the wind velocity was 0 ms - and max. eddy size was 5 0 cm, 000 eddies wold be sampled in abot 0 min. (b) In the PBL, similar reasoning leads to δz 0 3 cm, and the trblent elements shold be no larger than abot 0 m. Larger eddies are freqently encontered in the PBL. Since they can have very different characteristics, each averaging scheme mst be tailored to the eddy spectrm. These nmbers show that the eddy-continm hypothesis is a borderline assmption in many cases. Each flow sitation mst be careflly evalated with respect to the definition of a continm..9.3 Averaging The analysis of trblent flow is bilt on the definition of the mean. First, a mean mst be defined. Then trblence can be considered as a departre from the mean. In this section we will denote space averages with <( )> and time averages with ( ). An average over a spatial volme V of any physical parameter M can be written < M > = MdV (.6) V V Consider M to be a fnction of velocity, and vary over the space of the domain. So the average mst depend on. The variation can be handled with a probability density fnction (PDF), F{ (x, t)}. The PDF is defined by looking at the statistical average of a large nmber of flow samples. The probability is defined as -3

24 The nmber of times an event occrs Probability The total nmber of events observed When the event is a continos variable, like the wind speed, we define the PDF of i as The nmber of times that i i PDF The nmber of observations The average may then be written with the PDF as a weight factor in the averaging integral: < M{ (x, t)}> = PDF { (x, t)}m{ (x, t)}dv (.7) V This is a statistical or probability average, called an ensemble. Note that when the flow parameter is distribted niformly over space, it is called homogeneos. When the distribtion is niform with respect to time, it is called stationary. The attribtes of a statistically stationary flow over a homogeneos domain allow the simple mean vale with respect to time to be sed in place of the more rigoros statistical average. If one assmes that the characteristics of homogeneity and time dependence are sch that the time average is M = M x t dt t, where t = t t (.8) t lim (, ) t 0 t then this average arises from the ergodic hypothesis. Once the mean has been determined, the flctations M can be calclated from this mean: M = M M where M = 0 Figre.3 A sketch of a typical vale of velocity obtained from a boy that measres at 0 times per second when it is averaged for the interval shown..0 The Eqation of State for a Perfect Gas The ideal gas law for air can be written as p = ρrt (.4) where p pressre (Pa); ρ density (kg m -3 ); R specific gas constant ( cm s - o K - ); T temperatre ( o K). -4

25 The specific enthalpy (h) and the specific internal energy (e) can be related by this formla h = e p / ρ Note that the change of internal energy and the change of enthalpy can be given by e e = c v (T T ) (.5) h h = c p (T T ) (.6) where c v is the specific heat at constant volme c p is the specific heat at constant pressre Therefore, the specific heats can be defined in differential form, or in integral form, Note that e e = e ( ) T v c v, h ( ) T p c p T cdt T v, h h = T T cdt p h h = e e ( ρ p ) ( ρ p ) Hence = c v (T T ) R(T T ) (.7) c p = c v R (.8) The specific entropy can be written as S S = dq rev (.9) T Here, the heat transfer q takes place in a reversible process between state and state. Real processes always contain some irreversibility, therefore entropy will always increase. The entropy for an ideal gas can be related to the other state variables by this formla: T p S S = c p ln( ) Rln( ) (.a) T p or T ρ S S = c v ln( ) Rln( ) (.b) T ρ The most common process in the atmosphere and ocean is adiabatic, when -5

26 dq = 0. A process which is both adiabatic and reversible is called isentropic. From (.a) with S = S, we have T p = ( ) R/cp (.3) T p Water vapor behaves like a perfect gas so that the eqation of state for moist air (dry air and water vapor) can be written as R * R * R * p = ρ T = ρa T ρw T (.7a) m a m a m w where R* is the niversal gas constant ρ is the density of moist air (ρ = ρ a ρ w ) ρ a is the density of dry air ρ w is the density of water vapor m a is the moleclar weight of dry air m w is the moleclar weight of water Therefore eqation (.7a) can be re-written as m a ρ p = ρrt{ ( ) w } mw ρa ρ w m a = ρrt{ ( ) qh } (.7b) m w ρ w where q h = is specific hmidity ρa ρ w If we define the virtal temperatre as m a T v T{ ( ) qn } (.8) mw Then the eqation of state for moist air can be written as p = ρrt v (.9) For large-scale motion in the atmosphere and ocean, the flow is often assmed to be two-dimensional. With this approximation, the vertical momentm eqation is redced to the hydrostatic eqation as dp = ρg (.30) dz If a thermodynamic process has no net heat change or transfer, this process is called adiabatic. The adiabatic change of temperatre with height can be obtained from the first law of thermodynamics, dq = c v dt pd( ρ ) = cp dt ( ρ )dp (.3) where q is the rate of external heat added to the system. -6

27 In the adiabatic case, dq = 0, and from (.3) we know that dp ( )adiabatic = ρc p dt and from the hydrostatic eqation (.30), we can derive dt dp g ( )adiabatic = = Γad (.3) dz ρc p dz c p where Γ ad is called the adiabatic lapse rate. From eqation (.3) with dq = 0, dp c p dt = p (.33) This may be written as c p T dt = dp ρ T = R dp p Integrating the above eqation form p = 000 mb to p and from T = θ to T, we yield the potential temperatre eqation 000 θ T( ) R/cp (.34) p With the definition of potential temperatre in (.34),we can re-write the first law of thermodynamics (.3) as or T dq = c p dθ = cp Tdlnθ θ ds = c p dlnθ (.37a) (.37b) Therefore, for adiabatic process, there is no change of entropy and the potential temperatre is conserved.. Viscosity LIQUID GAS Figre.33 Sketch of a liqid and a gas moleclar transport at an imaginary plane marked by AA in the flid. Liqid: () many collisions, () intermoleclar forces exist across AA. Gas: () Relatively few collisions, () No intermoleclar -7

28 forces exist across AA. Figre.34 Sketch of the exchange of two molecles designated and across an imaginary srface AA. Figre.35 Sketch showing molecles of a flid at a solid bondary, on the scale of a mean free path. If there is a mean velocity U, then a mean stress τ exists. The moleclar momentm exchange is the process in a flid that prodces the internal stress force. It reqires a mean velocity shear to set p a difference in momentm pls random moleclar motion in the z-direction to move the momentm. The viscosity is simply a proportionality factor that represents the effectiveness of the moleclar exchange process. It empirically relates stress to mean shear. Viscosity from moleclar theory λ Moleclar flx across AA v The averaging molecle crossing AA begins at (/3)λ away The shear stress= µ Viscosity depends on temperatre and moleclar characteristics of the flid Parameterization Figre.36 The linear velocity shear in a -D parallel flow between plates -8

29 separated by h. The top plate is moved along by force F at velocity U. Consider a flow field in Fig..36, there stress τ can be described by Newton s law of friction: d τ = µ dz (.40) where the proportionality factor µ is defined as viscosity (or dynamic viscosity). µ The kinematic viscosity ν is defined as ν =. ρ Figre.37 Plot of stress verss rate-of-strain relations for varios categories of flid. In 877, Bossinesq introdced a mixing coefficient, or eddy viscosity, K, in an analogy with the laminar flow relation between the stress τ and the velocity shear so that d τ = ρk dz (.4) This assmes that the transport of properties (heat, momentm, etc) is done by trblent elements with scales mch smaller than that of the basic mean flow. Figre.38 Sketch of a bondary layer containing trblent eddies. In this case, assme that the eddies are generated by the srface roghness. -9

30 Chapter Flow Parameters. Local Time and Spatial Changes Lagrangian frame of reference: the reference frame is moving with the flid parcel of interest. Elerian frame of reference: the reference frame is fixed in the space. In the Largangian coordinate, the acceleration can be written as d ( t δt) ( t) = lim dt δt 0 δt However in the Elerian coordinate, the acceleration can be written as D ( x, t δt) ( x, t) ( x δ x, t) ( x, t) = lim δt, δx 0 δt δx Here, the capital D/ is sed to indicate a total derivative. Note that the first term on the right-hand side in the above eqation is the local time change, and the second term is the advective component de to the velocity gradient. Example.: Data taken in large-scale modeling is often gathered by ships traversing in a selected domain. A typical experiment might involve a stdy of the North Atlantic and a particlar ship that begins a rn at longitde 0 o W and latitde 40 o N in the fall. (See Fig...) As the ship proceeds de north, one of the parameters measred is the air temperatre at 800 Z each day for a 6-week period. The crise area and the data record are shown. Comment on the data trend. Discss the temperatre record measred at a ship anchored at P. Figre. A sketch of a ship (A) crising north past a fixed ship (B) with the daily temperatres of ship A. Soltion: The temperatre change measred by ship A is dt T T ( ) = ( ) ( ) dt -30

31 or ship A global motion dt T T ( ) = ( ) ( V ) where dt y ship A global motion dy V= dt The temperatre change measred by ship B (anchored) is dt T ( ) = ( ) dt ship B global An observer at point P can determine the time history of the temperatre on the moving ship only if the T(y) field and the ship speed V are known. For a flid property f(x, y, z, t), its total derivative can be determined from the chain rle as Df f f dx f = dt y f = t f x f v y dy dt f z f w z dx dy dz where the velocity components are (, v, w) = (,, ) dt dt dt Lagrangian vs. Elerian description dz dt (.5) Kinematics of Flid motion Z dx r r = ξ x r 0 Y r r r x = ξ dx X r A material element can be identified by its initial position x r ξ = 0 which is referred to as the material coordinate of that point. The locs of points x r traced ot by the element defines a material a parcel traectory, which is niqely determined by the material coordinate r ξ and the motion field v x r, t. ( ) -3

32 With r r r ξ held fixed, the vector dx = vdt describes the incremental displacement of the material element dring the time interval dt. The element s velocity is then r dx v = 3 dt Becase it is evalated for fixed r ξ, the time derivate in 3 corresponds to an individal material element and therefore to the Lagrangian time rate of change. By the kinematic constraint relating Elerian and Lagrangian descriptions, the v x r, t at the material element s velocity in 3 mst then eqal the field vale ( ) position x r () t at time t Streamline Streakline In addition to translation, a material element can ndergo rotation and deformation as it moves throgh the circlation. These effects are embodied in the velocity gradient r i V = i, =,,3 a -D tenser V r determines the relative motion between material coordinates, and hence the distortion experienced by the material element located at x r i i i = x x i x x i e w e i = rate of strain or deformation tensor, each diagonal component represents stretching effect r i e.g. eii = = V rate of increase of volme or dilatation i off diagonal component shear effect w i = vorticity The material Derivative i To transform to the Elerian descrption, the Lagrangian derivative appearing in conservation laws mst be expressed in terms of field property. Consider a field variable ψ = ψ ( x, y, z, t). The incremental change of property ψ is described by the total differential i -3

33 ψ ψ ψ ψ dψ = dx dy dz y z ψ r = dt ψ dx 4 Let dx r and dt denote increments of space and time with the material coordinate r ξ held fixed. dψ, the total differential, describes the incremental change of property ψ observed in a frame moving with the material element. The Advective Change and Index Notation In 3D space, a small differential of the velocity field ( x, t) may be written as δ = ( x δ x, t δt) ( x, t), where δ x = δt In the Cartesian coordinate, δx = δt, δy = vδt, δz = wδt, or δx i = i δt, i =,, 3 the index i can take any of three vales corresponding to the three-dimensionality of space. Using this index notation, the total velocity differential δ can be written as ( t δ t) ( t) δ = δt δt x x ( δ ) ( ) δx δx x ( x δx ) ( x ) ( 3 δ 3) ( 3) δx δx3 δx δx3 and it can be frther written as, δ ( i 3 k) ( x δx) ( x) δx δ = δt [ δt δt δx δt x x x So ( x δx ) ( x ) δx ( x3 δx3) ( x3) δx3 δt δt ] i δx δt δx δt ( x δx) ( x) δx ( x δx ) ( x ) δx [ δt δt δx δt δx δt 3 ( x3 δx3) 3( x3) δx3 δt ] [ ] k δx δt

34 δ δ δx δ δx δ δx3 δ = δt [ δt δt δt ] i δt δx δt δx δt δx δt δ δx δ δx δ3 δx3 [ δt δt δt ] [ ] k δx δt δx δt δx δt Let s divide the above eqation by δt, we have 3 3 δ δ = [ δ ( δt δt δ x δ δ x δ 3 δ x 3 )δt] i where δ [ δ ( δt δ x δ δ x δ 3 δ x 3 )δt] [ ] k δx i = i δ t Using the smmation convention, the repeated index () in a single term implies a sm over the range of index (=,, 3 in this case). Then the above eqation can be rewritten as δ i δt δ ={ i δ t δ δ x i } Where δ δ x i = δi δx when we take the limit, we have δ δx i 3 δ δx i 3 δi lim [ δt, δx 0 δt δ = i i δ, δt δx δ ] = δ δx i i = Di where D i = the i-component of acceleration Therefore, the above identity can be expressed in vector form as D = ( ) (.6) t total derivative local advective.3 Divergence/Convergence For a vector F, -34

35 F = F i F F3 k F = ( i k ) (F i F F3 k ) y z F = F F 3 Fi = y z i Note that the divergence of the vector F is a scalar. If the vector is the velocity vector, then is the divergence of the flid at a point. If the flid is incompressible, then the velocity divergence is zero ( =0) For a convergent flow field < 0 (e.g., sink) For a divergent flow field > 0 (e.g., sorce) Example.: Show that for Cartesian coordinates, Soltion: ( Φ) = Φ Φ (Φ is a scalar, is a vector) ( Φ) = [Φ( i 3 k )] =.4 Vortitity (Φ ) (Φ ) (Φ3 ) y z = Φ( Φ Φ y = Φ Φ 3 ) z Φ Φ Φ 3-35

36 Figre.6 (a) Cross-section of a hrricane. Horizontal extent is km; vertical extent is 0-5 km. Vertical wind vectors are -5 m/sec. (b) Two-dimensional windfield at the srface of a hrricane. Wind vectors are 0-90 m/sec. Vorticity is a physical concept associated with the rotation of the parcel at a local point. It is the cross prodct of del and the velocity vector. or in matrix form, ω = (.7a) i ω = y v k z w w v = ( ) w i ( ) v ( ) k y z z y (.7b) Example.3: What is the vorticity for flow velocity given by (a) = (3x,, 0), (b) = (Cy, 0, 0)? Soltion: -36

37 (a) The vorticity: v ω = = = y Note that the divergence is v w D = = y z (b) The vorticity ω = = (0, 0, -C) () (3x) = 0 0 = 0 y = (3x) = 3 For D horizontal flows, w = 0 and / z = 0, so that only the vertical component ζ of ω is nonzero. v ζ = ( ) k y In this case, the vertical vorticity is simply a scalar. Many important geophysical flows are basically horizontal, and the vorticity is a scalar field. This simplification makes it mch easier to nderstand and apply vorticity behavior. Vorticity characterizes the tendency of the parcel element to rotate abot its center. These, if a parcel-size wheel (like a water wheel) with paddles is placed in the flow, it will rotate with the parcel in proportion to the vorticity of the parcel. Figre.7 If it is very small with respect to the flow domain dimensions, a paddle wheel may approximate the spin of a parcel. Flow in two-dimensional shear gives rise to differential velocity across the axis of the paddle wheel, casing rotation. Example.4: Discss the vorticity for the two two-dimensional flows shown in Fig..8 and describe the motion of a vorticity meter placed in the flow. In the first, the channel is narrow, so that the bondary layer effects slowing the flow are significant. In the second, the bondary layers are negligibly thin. Assme for now that at a point P in the center of the crve the velocity can be -37

38 approximated with = 0 / C x C y v = C x C y Figre.8 Examples of parallel flow in a straight and a crved channel. The straight channel is narrow so that side effects are felt. The crved channel is wide so that side effects are negligible. Soltion: () For the narrow channel flow, ω = i = 0 y 0 k 0 0 = k y So a vorticity meter will experience torqe on the vanes extending in the y direction. The vorticity meter will rotate conter-clockwise above the centerline and clockwise below the centerline. Becase of symmetry, / y= 0 at the centerline. Ths, if the vorticity meter is placed precisely at the centerline, there is no vorticity. Figre.9 Velocities on the vane of a wheel (a vorticity meter) in the parabolic flow in a relatively narrow channel. () For the crving channel flow, at point p, -38

39 ω = i = y v k 0 0 v = ( ) k y 0 Since = C x C y v = C x C y 0 ω = [ ( C x C y) ( Cx C y)] k y = ( C C ) k = 0 So there is no vorticity in the center of the crve..5 The Vortex Vortex is a term describing a vorticity concept that has a specific vorticiy field associated with it. The concept of a vortex involves flid rotating arond a central axis. For the free vortex, the flow motion satisties this relation or Vr = constant = C (.8) C V = θ = r (.8a) Figre.0 The free vortex. Flow is tangential and inversely proportional to r. Example.5: Show that the vorticity is zero everywhere in a free vortex except at the origin. Soltion: Since a vortex geometry is a cylindrically rotating colmn of flid, the -39

40 eqation will be simplest when written in cylindrical coordinate (r, θ, z). For flow velocity Vorticity vector is or r = r e eθ z θ z e er ω = = r r r r eθ θ r θ e z z z z θ r z ( r θ ) = ( ) er ( ) eθ ( r ) e z r θ z z r r r θ Then for the free vortex, we have So for r > 0 r = z = 0, = [0, 0, r r C θ = r (C)] = [0, 0, 0] 0 At r = 0, the vorticity is indeterminate (Q ζ = ) 0 For an realistic flow sitation, the origin has an infinite velocity and it mst be exclded. For the forced vortex (solid-body rotation), the flow motion satisfies the relation or V = constant = Ω (.9) r V = rω (.9a) dθ where Ω =, the anglar rotation rate (or anglar velocity) dt Figre. The forced vortex. The flow is tangential and linearly proportional to r. One component of the earth s rotation will prodce an effective rotation of the earth s srface, which will in trn force a vortex flow from the srface of the -40

41 earth into the atmosphere. The rotation rate is one revoltion every 4 hors at the poles and zero at the eqator. Since the component of the earth s rotation changes sing at the eqator, the cyclones and hrricanes will rotate conter clockwise in the northern hemisphere and clockwise in the sothern hemisphere. Figre. The centripetal force/nit mass. The centripetal force is directed inward toward the axis of rotation. The gravitational force wold be directed along r to the center of mass of the earth. Example.6: Calclate the vorticity field for the velocity field given by dθ = (0, rω, 0), where Ω = = constant dt Soltion: The vorticity in the cylindrical coordinate is = r e r r r r eθ θ r θ e z z z For the given volecity field, r = z = 0, θ = rω = r e r r 0 r eθ θ r Ω e z z 0 = ( r Ω) e z = Ωe z r r -4

42 .6 The Coriolis Term When we apply the Newton s law, F = ma, in the earth-based frame of reference which is rotating and noninertial, a virtal force is reqired to be added. In geophysical flows, the virtal forces are centrifgal and Coriolis force. Initially we consider the acceleration with respect to a nonrotating coordinate system, fixed in the earth as shown in Fig... We can calclate the velocity as r r the derivative of the position vector ( V = dr/ dt) and the acceleration as the r r derivative of the velocity vector ( a= dv / dt). The derivative of a vector with respect to the absolte coordinate is related to the derivative in the rotating coordinate by the operator (see derivation below): d d ( ) A = ( Ω ) dt dt where Ω = ( 0, Ω E cosφ, Ω E sinφ) is the vector along the axis of rotation, Ω E is the rotation rate of the earth, and φ is the latitde. dr dr Ths, V A = ( ) A = Ω r = V VE (.0) dt dt where V is the relative velocity, V E is the velocity de to the earth s rotation, and V A is the absolte velocity with respect to the fixed coordinate. Similarly, dv d ( ) A = ( Ω )( V Ω r) dt dt dv dr = Ω ( Ω r) Ω V Ω dt dt dv dv ( ) A = ( ) r Ω ( Ω r) Ω V (.) dt dt Absolte Relative Centripetal Coriolis Acceleration Acceleration Acceleration Force Using the Newton s law, the absolte acceleration can be written as dv ( ) A = F dt m Therefore the relative acceleration in the rotating coordinate can be expressed as dv ( ) r = F Ω ( Ω r) Ω V dt m Relative Net force The virtal The Coriolis acceleration Acceleration per nit mass centrifgal force -4

43 d d r d d r Derivation of ( ) A = ( Ω ), or ( ) I = ( Ω ) dt dt dt dt Consider a vector A r which has constant magnitde bt which rotates with anglar velocity Ω r (like the nit vectors in a Cartesian coordinate, i r, i r, i r 3.). Let the angle between A r and Ω r be γ. In a small t, A r is rotated throgh the angle θ = Ω t, where Ω r is the magnitde of Ω r ( dγ / dt). It is apparent from the left figre that the small change in A r is given by Ω r r r r r r At ( t) At ( ) A= nasin γ θ ( θ) where n r is the nit vector in the direction of change of A r, which mst be perpendiclar to A r A r γ (since its length is fixed) and perpendiclar to Ω r (by the definition of the rotation). Ths r r r r r A r () t n =Ω A/ Ω A. In the limit t 0, r r v r A da r dθ Ω A lim = = A sin γ v r, bt since t 0 t dt dt Ω A v r v r Ω A = Ω Asinγ, we have finally r v r da/ dt =Ω A for a vector of fixed magnitde. An observer who is fixed in the rotating frame of reference wold see no change in A r, which an observer in a non-rotating frame wold see A r change as described by. Both observers wold see the same vector, since by definition a vector A r is independent of the coordinate frame and to describe it. However, there perception of the rate of change of A r change morbidly. Note thogh that since r r d A d r r r da r r r = A A = A = A ( Ω A) = 0, dt dt dt both observers agree that the magnitde of A r is nalted. Consider now an arbitrary vector B r and a coordinate frame of reference rotating with anglar velocity Ω r. For simplicity let the frame be Cartesian with nit vectors along each axis, i r, i r, i r 3. The vector B r may be reprented in this rotating frame as r r r r r B = B i B i B, where B = i B, =,, 3. 3i3 Ω r i r 3 B r i r i r -43

44 The time rate of change of B r for an observer fixed in a rotating frame is simply r db db r db r db r 3 = i k dt dt dt dt R for in this frame the nit vectors are fixed in length and direction. On the other hand, for a non-rotating observer both the components of B r and the nit vectors change with time. The rate of charge of the scalar components B, B, B3 are common to both observers, so r r r r db db r db r db r 3 di di di3 i k3 B B B3 dt = 3 dt dt dt dt dt dt I where the sbscript I denotes rate of charge as seen by the observer in the non-rotating, inertial frame. r di B B i dt = Ω r r or Ω r B r so that 3 becomes r r db db r r d r d r r = Ω B dt dt, or ( ) I B = ( Ω ) B dt dt I R.6. Relative Vorticity In atmospheric and oceanic sciences, the rotating earth coordinate contribtes a component of planetary vorticity to the total vorticity, so that ζ t = ζ r ζ p (.) total relative planetary vorticity vorticity vorticity The component of planetary vorticity normal to the local srface (in the z-direction) is called to Coriolis parameter as defined by f = Ωsinφ (.3) For simplicity, the relative vorticity can be written st as ζ and therefore the absolte vorticity ζ t = ζ f.6. An Example of a Vorticity Calclation -44

45 Figre.3 An example of windfields sed to calclate streamline and vorticity fields. The wind fields are obtained from satellite data over the ocean srface. The vorticites are calclated for each grid point based on neighboring points..7 Integral Theorems.7. Leibnitz s Theorem The time rate of change of a conservative fnction f, integrated over a control volme that may be changing de to its motion with velocity, can be written as: D f fdv = dv f ( n) da (.5) V V () () wehre () = local change of f inside the control volme V () = the srface integral of the flx throgh the control srface S enclosing the control volme V. Note that Eq.(.5) is also called the Reynolds s Transport Theorem. The -D form of (.5) is the famos Leibnitz s theorem S D x= b( t) x= a( t ) f ( x, t) dx = b( t) a( t) f dx f ( b, t) db dt da f ( a, t) dt () () (3) The physical meanings of (), (), and (3) are shown in Fig

46 Figre.4 The parts of area nder f(x, t) represented in Leibnitz s theorem. The fnction is shown at times t and t dt. The total change in f incldes areas,, and 3 as points A, B, C and D move to A, B, C and D..7. Divergence Theorem V dv = ( n) da (.8) S () () where () = the volme integral at inside the control volme V () = the srface integral of the component of directled normal to the control srface S n = a nit vector otward from the control volme V at the srface area increment da Figre.5 Elements of a volme and its srface that are related in the Divergence Theorem. Example.7: Consider the steady nondivergent flow of an incompressible flid throgh a dct of constant area of 5 m at a section where there is a spigot feeding 0.-m 3 /sec flid into the flow (Fig..6). The flow velocity in is = 4 m/sec. What is the flow velocity ot? Use the Leibnitz theorem with f = ρ (the constant flid density). -46