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This book discusses the subject of wave/current flow around a cylinder, the forces induced on the cylinder by the flow, and the vibration pattern of slender. Request PDF on ResearchGate | On Jan 1, , B. Mutlu Sumer and others published Hydrodynamics Around Cylindrical Structures. 12 Hydrodynamics Around Cylindrical Structures by B Mutlu Sumer and Jorgen Fredsoe (Tech. Univ. of Denmark) Vol. 13 Water Wave Propagation Over.
Flow around a cylinder in steady current the opposite sign will t h e n cut off further supply of vorticity to Vortex A from its boundary layer. T h e effect of current shear on vortex shedding. Therefore it is important to know what kind of changes take place in the flow around and in the forces on such a pipe. H Time Time Figure 3. Putzig, D. T h e latter figure also contains information about the oscillating lift force and the Strouhal number, which are maintained in the figure for t h e sake of completeness. As has been discussed in the context of t h e effect of roughness, the increased level of incoming turbulence will directly influence the cylinder boundary layer and hence its separation.
You already recently rated this item. Your rating has been recorded. Write a review Rate this item: Preview this item Preview this item. Hydrodynamics around cylindrical structures Author: Singapore ; River Edge, NJ: Advanced series on ocean engineering , v.
English View all editions and formats Rating: Subjects Offshore structures -- Hydrodynamics. Underwater pipelines. Cylinders -- Hydrodynamics. View all subjects More like this Similar Items. Find a copy online Links to this item Knovel. Allow this favorite library to be seen by others Keep this favorite library private.
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Freds0e and Hansen's modified potential-flow solution, as is seen from Fig. W h e n a shear is introduced in the approaching flow, the variation of the lift force with respect to the gap ratio changes considerably very close to the wall, as seen in Fig. T h e shear-flow d a t a plotted in this figure were obtained in an experiment conducted at practically the same Reynolds number, employing the same test cylinder as in Fig.
T h e only difference between the two tests is t h a t in the shear-free flow experiments the cylinder was towed in still water, while in the shear-flow experiments the cylinder was kept stationary and subject to the boundary-layer flow established in an open channel with a smooth b o t t o m. Clearly, the difference observed in Fig.
T h e lift undergoes a substantial drop for very small gap ratios. Freds0e and Hansen links this drop to t h e change in the stagnation pressure in the following way: First they show t h a t t h e stagnation point does not move significantly by t h e introduction of the shear.
So the direction of pressure force is much t h e same in b o t h cases. T h e major difference is t h a t t h e stagnation pressure is reduced considerably with the introduction of the shear, Forces on a cylinder near a wall u FL dh i 4. Experiments, 10 as sketched in Fig. Clearly, the pressure in Eq. This reduction in the stagnation pressure, while keeping the direction of 62 Chapter 2: The stagnation point moves to lower and lower angular positions, and the suction on the free-stream side of the cylinder becomes larger and larger than that on the wall side.
Stagnation pressure decreases considerably in the shear-flow case. W h e n t h e cylinder is moved extremely close to t h e wall, however, more a n d more fluid will be diverted to pass over the cylinder, which will lead to larger and larger suction pressure on the free-stream side of the cylinder.
Indeed, when the cylinder is sitting on t h e wall, the suction pressure on t h e cylinder surface will be the largest Fig. This effect may restore t h e lift force in the shear-flow case for very small gap values, as is implied by Fig.
Although the shedding exists for gap ratios larger t h a n 0. Therefore t h e oscillating forces will be affected, too, by t h e close proximity of t h e wall.
T h e figure shows t h a t t h e oscillating lift becomes weaker and weaker, as the gap ratio is decreased. Note t h a t t h e C'L coefficient here is defined in t h e same way as in Eq.
T h e CL coefficient plotted in the figure representing t h e vortex-induced oscillating lift is the lift coefficient associated with the m a x i m u m value of the oscillating lift force. As is seen from t h e figure, the wall-induced lift and t h e vortex-induced lift appear t o be in t h e same order of 64 Chapter 2: Forces on a cylinder in steady current II Freds0e etal.
T h e results regarding the flow description have been given in Section 1. T h e force coefficients are defined, based on the undisturbed velocity at the axis of the pipe. As mentioned in the flow Forces on a cylinder near a wall 65 7: It is interesting to note that CQ a n d CL reach their equilibrium values at rather early stages of t h e scour process. It is also interesting to observe t h a t the pipe experiences a negative lift force as soon as the tunnel erosion Stage II comes into action.
It is seen t h a t this lift force remains negative throughout the scour process. As for Stage V, t h e negative lift can be explained by t h e position of t h e stagnation point a n d t h e angle of attack of the approaching flow. This angle can in Fig. T h e phenomenon, namely the " p r e m a t u r e " vortex shedding, which causes t h e high Strouhal numbers in the initial stages of the scour process Stages III and IV in Fig.
Since the vortices shed from the pipe become stronger and stronger as t h e scour progresses, the fluctuating lift force should 66 Chapter 2: The coefficient CL is based on the amplitude of the oscillating lift force.
T h e plane-bed counterpart of each scour profile is selected on the basis of equal non-dimensional clearance between the pipe and the bed i. As for the mean lift coefficient CL, the difference between a plane bed and a scoured bed is t h a t t h e pipe experiences a negative lift force in the case of a scoured bed, while it experiences a positive one when the bed is plane Fig.
Stansby and Starr report the results of measurements of drag on a pipe undergoing a gradual sinking, as t h e scour process progresses in a live, sand bed. This is obviously due to the fact t h a t the pipe is protected against the flow, as it is buried in the sand bed. Forces on a cylinder in steady current a 1. W h e n the pipelines are placed in a trench hole, the forces are reduced considerably Fig.
As seen, both the drag and the lift are reduced by a factor , depending on the position of the pipe in t h e trench hole. This is because the pipe is protected against the main body of the flow by the trench sheltering effect.
Jensen and Mogensen report t h a t in the case of a trench hole the same size as t h a t in Fig. Forces on a cylinder near a wall Trench 1: Jensen and Mogensen Achenbach, E. Influence of surface roughness on the cross-flow around a circular cylinder. Fluid Meek, On vortex shedding from smooth and rough cylinders in the range of Reynolds numbers 6 X to 5 x 10 6.
Bursnall, W. Drescher, H.
Messung der auf querangestromte Zylinder ausgeiibten zeitlich veranderten Driicke. Flugwiss, 4 Freds0e, J. Lift forces on pipelines in steady flow.
Transverse vibrations of a cylinder very close to a plane wall. Offshore Mechanics and Arctic Engineering, Giiven, O. Surface roughness effects on the mean flow past circular cylinders. Iowa Inst. Hydraulic Res. A model for high-Reynolds-number flow past rough-walled circular cylinders. References 71 Giiven, O. Surface-roughness effects on the mean flow past circular cylinders. Hallam, M. Dynamics of Marine Structures. Hoerner, S. Fluid-Dynamic Drag. Published by t h e Author.
Obtainable from ISVA. Jensen, R. Hydrodynamic forces on pipelines placed in a trench under steady current conditions. Jones, W. Forces on submarine pipelines from steady currents. Study on the turbulent shear flow past a circular cylinder. Bulletin Faculty of Engrg. Kozakiewicz, A. Forces on pipelines in oblique attack.
Steady current and waves. Offshore and Polar Engineering Conf. Miiller, W. Systeme von Doppelquellen in der ebenen Stromung, insbesondere die Stromung u m zwei Kreiszylinder. Zeitschrift fur angewandte M a t h e m a t i k und Mechanik, 9 3: Turbulence and Reynolds number effects on t h e flow and fluid forces on a single cylinder in cross flow.
Forces on a cylinder in steady current Norton, D. W i n d tests of inclined circular cylinders. Parkinson, G. On the aeroelastic instability of bluff cylinders. Flow forces on a cylinder near a wall or near another cylinder. Research, Fort Collins, Co. Shih, W. Experiments on flow past rough circular cylinders at large Reynolds numbers. Stansby, P. On a horizontal cylinder resting on a sand bed under waves and current. Thorn, A. An investigation of fluid flow in two dimensions.
London, R. Thomschke, H. Yamamoto, T. Wave forces on cylinders near plane boundary. Waterway, Port, Coastal Ocean Div. Zdravkovich, M. Forces on a circular cylinder near a plane wall. Applied Ocean Research, 7: Chapter 3. Flow around a cylinder in oscillatory flows As shown in Chapter 1, the hydrodynamic quantities describing the flow around a smooth, circular cylinder in steady currents depend on the Reynolds number. In the case where the cylinder is exposed to an oscillatory flow an additional p a r a m e t e r - the so-called Keulegan-Carpenter number - appears.
T h e physical meaning of the KC number can probably be best explained by reference to Eq. T h e numerator on the right-hand-side of the equation is proportional to the stroke of the motion, namely 2a, while the denominator, the diameter of the cylinder D, represents t h e width of the cylinder Fig. Small KC numbers therefore mean t h a t t h e orbital motion of t h e water particles is small relative to the total width of the cylinder.
W h e n KC is very small, separation behind the cylinder may not even occur. Figure 3. Large KC numbers, on the other hand, mean that the water particles travel quite large distances relative to the total width of the cylinder, resulting in separation and probably vortex shedding. Creeping laminar flow. Separation with Honji vortices. S e e Figs. Turbulence over t h e cylinder surface A. Source for KC 4 Williamson Limits of the KC intervals may change as a function of Re see Figs.
Flow around a cylinder in oscillatory flows ranges. We shall concentrate our attention first on the KC dependence, however. T h e influence of Re will be discussed in Section 3. T h e separation first appears when KC is increased to 1. W h e n this KC number is reached, the purely two-dimensional flow over the cylinder surface breaks into a three-dimensional flow p a t t e r n where equally-spaced, regular streaks are formed over t h e cylinder surface, as sketched in Fig.
These streaks can be made visible by flow-visualization techniques. Observations show t h a t the marked fluid particles, which were originally on the surface of t h e cylinder, would always end up in these narrow, streaky flow zones.
T h e observations also show t h a t these streaks eventually are subject to separation in every half period prior to the flow reversal, each separated streak being in the form of a mushroom-shape vortex Figs.
This phenomenon was first reported by Honji and later by Sarpkaya a. Subsequently, Hall carried out a linear stability analysis and showed t h a t t h e oscillatory viscous flow becomes unstable t o axially periodic vortices i. Vortices B in Fig.
Flow regimes as a function of KC number 77 Figure 3. Oscillatory flow is in the direction perpendicular to this page. From Honji with permission - see Credits.
T h e flow regime where separation takes place in the form of Honji instability occurs in a narrow KC interval, namely 1. T h e following section will focus on these flow regimes. These works have shed considerable light on the understanding of the complex behaviour of vortex motions in various regimes.
Based on the previous research and his own work, Williamson has described the vortex trajectory patterns in quite a systematic manner. T h e following description is mainly based on Williamson In the vortex-shedding regimes the vortex shedding occurs during the course of each half period of t h e oscillatory motion.
Flow around a cylinder in oscillatory flows KC number. These KC ranges are 7 7 Figure 3. In this photograph the cylinder is moving up, and is near the end of a half cycle. In this case the street travels to the right. Williamson with permission see Credits. W h e n the flow reverses Fig.
As the half period progresses, Vortex M itself is shed and, being a free vortex, it forms a vortex pair with Vortex N Fig. As implied in the preceding, the concept "pairing" here means t h a t two vortices, of opposite sign, come together and each is convected by the velocity field of the other. It is evident from the figure t h a t there will be one vortex pair convecting away from t h e cylinder at t h e end of each full period.
Observations show, however, t h a t the vortex street changes sides occasionally. T h e position of the vortex street relative to the cylinder may be important from the point of view of the lift force acting on the cylinder. Due to the asymmetry, a non-zero mean lift must exist in this flow regime.
W h e n the vortex street changes side, then the direction of this lift force will change correspondingly. Flow around a cylinder in oscillatory flows occurs at one side only. From both Fig. Consider the oscillatory flow given in Fig.
From Eqs. As is seen, this is nothing but the Strouhal law with the normalized frequency being 0. So, as a conclusion it may be stated t h a t the observed increase in the number of vortices shed, namely 2 in one full period when KC range is changed to a higher regime, is a direct consequence of the familiar Strouhal law.
V o r t e x - s h e d d i n g f r e q u e n c y a n d lift f r e q u e n c y In contrast to steady currents, the concept "frequency of vortex shedding" is not quite straightforward in oscillatory flows, particularly for lower KC regimes such as the single-pair regime and the double-pair regime.
This is mainly due to the presence of flow reversals. T h e subject can probably be best explained by reference to Figs. These figures depict time series of the lift force acting on a cylinder and the corresponding motion of vortices, which are reproduced from Figs. T h e force time series have been obtained simultaneously with the flow visualizations of vortex motions so t h a t a direct relation between the lift variation and the motion of vortices could be established, Williamson Lift-force time series obtained simultaneously in the same experiment as the flow visualization study of vortex motions depicted in Fig.
The vertical arrows refer to cylinder motion. In the lift-force time series, the peaks marked A and C are caused by the growth and shedding of Vortex N and M Frames 1 and 4 respectively, while the peak marked B is caused by the return of Vortex N towards the cylinder just after the flow reversal Frame 3.
Flow around a cylinder in oscillatory flows discussion, Section 2. T h e positive peaks, on the other hand, for example t h a t marked B are induced by the r e t u r n of the most recently shed vortex towards the cylinder just after flow reversal such as N in Frame 3. The fact t h a t the cylinder experiences a positive lift force when there is a vortex moving over the cylinder in the fashion as in Frame 3 was shown also by the theoretical work of Maull and Milliner As is seen, not all the peaks in the lift force time series are induced by the vortex shedding.
H Time Time Figure 3. The quantity a is the variance of the lift fluctuations. Justesen As a rule, we may say that the peak in the lift force which occurs just after the flow reversal is related to the return of the most recently shed vortex to the cylinder, while the rest of the peaks in the lift variation is associated with the vortex shedding.
So, it is evident t h a t , in oscillatory flows, t h e lift-force frequency is not identical to the vortex-shedding frequency. One way of determining the lift frequency is to obtain the power spectrum of the lift force and identify the dominant frequency. As seen, the fundamental lift frequency normalized by the oscillatory-flow frequency, namely 88 Chapter S: Flow around a cylinder in oscillatory NL flows Ik 3.
This slight difference with regard to the KC number is related to the Reynolds number dependence. Table 3. Flow around a cylinder in oscillatory flows turbulence RexlO" Figure 3. The diagram is adapted from Sarpkaya a.
This is linked with t h e transition to turbulence in the b o u n d a r y layer. Once the flow in t h e b o u n d a r y layer becomes turbulent, this will delay separation and therefore the non-separated flow regime will be re-established. However, in this case, t h e non-separated flow will be no longer a purely viscous, creeping type of flow, b u t rather a non-separated flow with turbulence over t h e cylinder surface.
T h e transition to separated flow, on the other hand, occurs directly with t h e formation of a pair of symmetric vortices Region d, in Fig.
Lines, Sarpkaya a and Williamson and; squares from Justesen The quantity NL is the number of oscillations in the lift force per flow cycle: It is evident that no detailed account of various upper Reynolds-number regimes, known from the steady-current research such as the lower transition, the supercritical, the upper transition a n d t h e transcritical regimes , is existent.
Nevertheless, Sarpkaya's 92 Chapter 3: Flow around a cylinder in oscillatory flows a extensive d a t a covering a wide range of KC for lower Re regimes along with Williamson's and Justesen's d a t a may indicate what happens with increasing the Reynolds number. Regarding the vortex-shedding regimes, it is evident from t h e figure that the curves begin to bend down, as Re approaches to t h e value 10 5 , meaning that in this region t h e normalized lift frequency Ni increases with increasing Re.
This is consistent with the corresponding result in steady currents, namely t h a t the shedding frequency increases with increasing Re at 3. As has been seen, several changes occur in the flow around t h e cylinder when the cylinder is placed near a wall, such as the break-up of symmetry in t h e flow, the suppression of vortex shedding, etc.. T h e purpose of the present section is to examine the effect of wall proximity on the regimes of flow around a cylinder exposed to an oscillatory flow.
T h e analysis is mainly based on the work of Sumer, Jensen and Freds0e where a flow visualization study of vortex motions around a smooth cylinder was carried out along with force measurements. Re-range of the flow-visualization experiments was 10 3 —10 4 , while t h a t of the force measurements was 0. Gap-to-diameter-ratio values: Sumer et al. Jacobsen, B r y n d u m and Freds0e give a detailed account of the latter where the motion of the lee-wake vortex over the cylinder is linked to the maximum pressure gradient in the outer flow.
T h e liftforce trace is presented in Fig. Velocity, U t 0 Figure 3. Effect of wall proximity on flow regimes 95 96 Chapter 3: L-l ' M Figure 3.
Likewise, Fig. T h e reason why vortex shedding is maintained for such small gap ratios is because the water discharge at the wall side of the cylinder is much larger in oscillatory flow at small KC t h a n in steady currents due to the large pressure gradient from the wave.
Open symbols: Filled symbols: Flow around a cylinder in oscillatory flows T h e frequency of vortex shedding can be defined by an average frequency based on the number of the short-duration peaks in the lift force over a certain period, as sketched in Fig. T h e shedding frequency actually varies over the cycle.
Also plotted in Fig. T h e details regarding these two latter studies have already been mentioned in the previous chapter see Fig. From Figs. This is because t h e presence of the wall causes the wall-side vortex to be formed closer to t h e free-stream-side vortex.
As a result of this, the two vortices interact at a faster rate, leading to a higher St frequency. Finally, Sumer et al. T h e studies concerning the effect of Re number, the effect of cylinder vibration, and the effect of turbulence in the incoming flow on correlation in steady currents have been reviewed in Section 1. In the present section, we will focus on the correlation measurements m a d e for cylinders exposed to oscillatory flows.
Correlation 0. Obasaju et al. In t h e study of Obasaju et al. T h e correlation coefficient is calculated from the signals received from the pressure transducers mounted along the length of the cylinder using t h e following equations, Eqs. T h e overbar in t h e preceding equations denotes ensemble averaging: This phase value corresponds to the instant where the flow at the measurement points comes to a standstill, as can be traced from the pressure traces given in Kozakiewicz et al.
As the flow progresses from this point onwards, however, the correlation gradually decreases and assumes its Chapter S: T h e n it increases again towards the end of t h e next half period. T h e flow picture in Fig. As time progresses from this point onwards, however, this vortex begins to move in the reverse direction and is washed over the cylinder as a coherent entity along t h e length of the cylinder Fig. Now, comparison of Fig. This is indeed the case found in the preceding in relation to Fig.
The cylinder is located well away from a wall, namely the gap-t therefore, the effect of wall proximity could be considered insignificant. Wall proximity effect regarding the pressure fluctuations. See Fig. Correlation T h e correlation coefficients presented in Fig. However, caution must be exercised in interpreting the results in the figure. However, the lift in this case consists of two p a r t s , a low frequency portion which is caused by the close proximity of the wall and the superimposed high-frequency fluctuations which are caused by vortex shedding Fig.
Regarding the correlation of the lift force itself, Kozakiewicz et al. Clearly, with this arrangement t h e pressure time-series can be substituted in place of t h e lift force ones, as far as t h e correlation calculations are concerned.
Hence, the correlation in connection with t h e lift force in this case cannot be calculated by Eq. T h e correlations in these diagrams are now all associated with the lift force; therefore comparison can be m a d e on t h e same basis. T h e figure indicates t h a t , as expected, t h e correlation increases tremendously as the gap ratio changes from 2. Rx for the wall-mounted cylinder is computed direct from pressure signals employing time-averaging according to Eq.
Correlation Effect of v i b r a t i o n s o n c o r r e l a t i o n This section focuses on the effect of vibrations on the correlation when the cylinder is vibrated in a direction perpendicular to the flow only.
In the study of Kozakiewicz et al. Also, the cylinder vibrations were synchronized with t h e outer oscillatory-flow motion.
T h e results of Fig. Note t h a t in Novak and Tanaka's study t h e cylinder is vibrated with a frequency equal to its vortex-shedding frequency, which is identical to the fundamental lift frequency. Likewise, in t h e study presented in Fig. In Kozakiewicz et al. Note t h a t these figures are in accordance with Sarpkaya's a stationary-cylinder lift-force frequency results Fig. Returning t o Fig. This can be seen even more clearly from Fig.
T h e way in which the correlation coefficient increases with increasing amplitude-to-diameter ratio is in accord with the steady current results Fig. However, this increase is not as large as in steady currents. In a subsequent study, Sumer et al. Their results indicated t h a t the correlation increases monotonously with increasing amplitude of vibrations Fig.
T h e observed Ill Chapter S: Vibrations are forced vibrations and Ni being the normalized fundamental lift frequences Eq. Correlation Cylinder with forced vibrations.. Period-averaged correlation cofficient with respect to vibration amplitudes for different KC numbers; d: Vibrations are forced vibrations. In t h e tests of Kozakiewicz et al. In the tests of Sumer et al.
This may explain the disagreement between t h e results of Kozakiewicz et al. Vibrations are not forced, but rather selfinduced vibrations. As is seen, R decreases as the vibration frequency moves away from the fundamental lift-force frequency.
This result is in agreement with the corresponding result obtained in Toebes' study for the steady-current situation. This is called streaming. A simple explanation for the emergence of this steady streaming may be given as follows.
T h e flow velocity experienced at any point near the surface of the cylinder Point A, say, in Fig. Namely, the velocity is relatively larger when the flow is in the direction of converging surface geometry t h a n t h a t when the flow is in the opposite direction, as sketched in Fig. Flow around a cylinder in oscillatory flows Figure 3. Flow Flow from a t o b from b t o a Figure 3. This presumably leads to the recirculating flow p a t t e r n shown in Fig.
Streaming T h e streaming has been the subject of an extensive research with regard to its application in the field of acoustics see Schlichting , p. It may be i m p o r t a n t also in the field of offshore engineering in conjunction with the sediment motion and t h e related deposition and scour processes around very large, bottom-seated marine structures which are subject to waves.
The steady streaming caused by an oscillating circular cylinder. The thickness of recirculating cells, o, experiment, Holtsmark et a]. Wang developed an analytical theory for very small Re numbers creeping flow and KC numbers. Wang's results compare very well with the experiments. In t h e study, analytical expressions were obtained for t h e stream function and t h e drag coefficient.
For large Re numbers, apparently no study is available in t h e literature. Therefore it is difficult to make an assessment of the thickness of t h e recirculating cells and the magnitude of the streaming.
T h e timeaveraged flow field over one period obtained by the authors revealed t h e presence of the steady streaming p a t t e r n depicted in Fig. Viscous oscillatory flow about a circular cylinder at small to moderate Strouhal number.
Vortex trajectories in oscillatory flow. Hydrodynamic forces on cylindrical bodies in oscillatory flow. T h e role of vortices in oscillatory flow about bluff cylinders. Forces on cylinders in harmonically oscillating flow.
References Grass, A. Flow visualization studies of oscillatory flow past smooth and rough circular cylinders. T h e influence of b o u n d a r y layer velocity gradients and b e d proximity on vortex shedding from free spanning pipelines. Energy Resour. Hall, P. On t h e stability of the unsteady boundary layer on a cylinder oscillating transversely in a viscous fluid.
Holtsmark, J. Boundary layer flow near a cylindrical obstacle in an oscillating incompressible fluid. Honji, H. Streaked flow around an oscillating circular cylinder. Howell, J. Vortex shedding from a circular cylinder in turbulent flow. Jacobsen, V. Determination of flow kinematics close to marine pipelines and their use in stability calculations. In Proc. Paper O T C Justesen, P. Hydrodynamic forces on large cylinders in oscillatory flow.
Spanwise correlation on a vibrating cylinder near a wall in oscillatory flows. Fluids and Structures, 6: Sinusoidal flow past a circular cylinder.
Coastal Engineering, 2: Press, pp. Obasaju, E. A study of forces, circulation and vortex patterns around a circular cylinder in oscillating flow. Flow around a cylinder in oscillatory flows Ramberg, S. Velocity correlation and vortex spacing in t h e wake of a vibrating cable.
Sarpkaya, T. In-line and transverse forces on smooth and sand-roughened cylinders in oscillatory flow at high Reynolds numbers. Force on a circular cylinder in viscous oscillatory flow at low Keulegan-Carpenter numbers. Schlichting, H. Boundary-Layer Theory. Singh, S. Forces on bodies in oscillatory flow. Sumer, B. Transverse vibrations of an elastically mounted cylinder exposed to an oscillating flow.
Effect of a plane boundary on oscillatory flow around a circular cylinder. A note on spanwise correlation on a freely vibrating cylinder in oscillatory flow. Fluids and Structures, 8: Wave b o u n d a r y layers in a convergent tunnel.
Tatsumo, M. A visual study of the flow around an oscillating circular cylinder at low Keulegan-Carpenter numbers and low Stokes numbers. Wang, C. On high-frequency oscillatory viscous flows. Sinusoidal flow relative to circular cylinders. Vortex formation in t h e wake of an oscillating cylinder. Jour, of Fluids and Structures, 2: Chapter 4. Forces on a cylinder in regular waves Similar to steady currents, a cylinder subject to an oscillatory flow may experience two kinds offerees: In the following, first, t h e in-line force on a smooth, circular cylinder will be considered and subsequently the attention will be directed to the lift force.
T h e remainder of the chapter will focus on t h e influence on the force components of t h e following effects: Note t h a t t h e velocity-squared term in Eq. In t h e case of oscillatory flows, however, there will be two additional contributions to the total in-line force: T h e following paragraphs give a detailed account of these two forces.
Suppose t h a t a thin, infinitely long plate with the width b is immersed in still water and t h a t it is impulsively moved from rest Fig. W h e n the plate is moved in its own plane, it will experience almost no resistance, considering that the frictional effects are negligible due to the very small thickness of the plate.
Whereas, when it is moved in a direction perpendicular to its plane, there will be a tremendous resistance against the movement. T h e reason why this resistance is so large is t h a t it is not only the plate but also the fluid in the immediate neighbourhood of t h e plate, which has to be accelerated in this case due to t h e pressure from the plate.
T h e hydrodynamic mass is defined as the mass of the fluid around the body which is accelerated with the movement of the body due to the action of pressure. Usually, the hydrodynamic mass is calculated by neglecting frictional effects, i. Hereby the flow field introduced by accelerating t h e body through the fluid can be calculated using potential flow theory.
T h e procedure to calculate the hydrodynamic mass for a b o d y placed in a still water can now be summarized as follows.
In the following we shall implement this procedure to calculate the hydrodynamic mass for a free circular cylinder.
Forces on a cylinder in regular waves Example 4. W h e n a cylinder is held stationary and the fluid moves with a velocity U in the negative direction of the i-axis, the velocity potential is given by MilneThomson, , Section 6. This term, as a matter of fact, is not significant as it does not contribute to the resulting force. Cm for a circular cylinder is Eq. H y d r o d y n a m i c m a s s for a circular c y l i n d e r n e a r a w a l l W h e n the cylinder is placed near a wall the pipeline problem , the hydrodynamic mass will obviously be influenced by the close proximity of the wall.
Yamamoto et al. Their result is reproduced in Fig. As is seen, t h e hydrodynamic-mass coefficient Cm increases with decreasing t h e gap between the cylinder and the bed. Finally, it m a y be mentioned that simple algorithms for calculating hydrodynamic mass for cylinders placed near an arbitrarily shaped scoured sea bed were given by Hansen Hansen's calculations cover also groups of cylinders.
A number of examples including multiple riser configurations were given also in Jacobsen and Hansen In-line force in oscillatory flow J I I L 3e 2. This force is caused by the acceleration of the fluid in the immediate surroundings of the body.
W h e n the body is held stationary and the water is moved with an acceleration a, however, there will be two effects. First, the water will be accelerated in t h e immediate neighbourhood of t h e body in the same way as in the previous analysis. Therefore, the previously mentioned hydrodynamic mass will be present. This pressure gradient in t u r n will produce an additional force on the cylinder, which is termed t h e F r o u d e - K r y l o v force.
T h e force on the body due to this pressure gradient can be calculated by the following integration: ISO Chapter J: Prom the Gauss theorem, Eq. T h e total force, F, is given b y Eq. In the case when t h e b o d y moves relative to the flow in the in-line direction this may occur, for example, when the body is flexibly mounted t h e Morison equation, from Eq.
This phase difference should be taken into consideration if the maximum value of the in-line force is of interest. T h e ratio between the m a x i m u m values of the two forces, on the other hand, can be written from Eq. However, as t h e KC number is increased, t h e separation begins to occur Fig. As a rough guide we may consider the range of the Keulegan-Carpenter number 0 20 — 30 as the drag-dominated regime.
Finally, it may be mentioned t h a t , in some cases such as in t h e calculation of damping forces for resonant structural vibrations, t h e drag force becomes so important t h a t even the small contribution to t h e total force must be taken into consideration. This can be obtained by solving Laplace's equation. Boundary layer Figure 4. Milne-Thomson, , Section 6. In the case when KC 4. Forces on a cylinder in regular waves and n being the local coordinate Fig.
This b o u n d a r y layer will p e r t u r b the previously predicted potential-flow force in the following two ways: The friction force: Inserting Eq. T h e aforementioned effect can be considered as a source with the strength m determined from t h e following equation see Milne-Thomson, , Sections 8.
Using Eq. T h e results of the asymptotic theory given in the preceding paragraphs are t h e same as the Stokes' results to 0 [ i?
As is seen, the theory shows remarkable agreement with the experiments for very small values of KC where the flow remains attached cf. Experiments from Sarpkaya a. Asymptotic theory Eqs. T h e theory gives the explicit form of this dependence. However, this is for t h e combination of very small KC numbers and sufficiently large Re numbers only. Although there are several numerical codes developed to calculate flow around and forces on a cylinder in oscillatory flows Chapter 5 , these are still at the development stage and therefore not fully able to document the way in which the force coefficients vary with KC and Re.
Hence, t h e experiments appear to be the most reliable source of information with regard to the force coefficients at the present time. For periodic flows, the most suitable technique may be " t h e method of least squares". T h e principle idea of this method is t h a t the Co and CM coefficients are determined in such a way t h a t the mean-squared difference between the predicted by t h e Morison formula and t h e measured force is minimum.
A brief description of the method of least squares is given below. Let Fm t be the measured in-line force at any instant t. T h e first equation leads to: Solving for fd and fi, the in-line force coefficients Co and CM can b e determined from Eqs.
For a sinusoidal flow, it can be shown t h a t the method of least squares gives CD and CM as follows: Another technique regarding the experimental determination of Co and CM coefficients is the Fourier analysis. This latter technique yields identical CM values. As for Co, the Co values obtained by the Fourier analysis differ only slightly from those obtained by t h e method of least squares Sarpkaya and Isaacson, Keulegan and Carpenter were the first to determine t h e Co and CM coefficients for a cylinder exposed to real waves using the Fourier analysis.
Subsequently, Sarpkaya a m a d e an extensive s t u d y of t h e forces on cylinders exposed to sinusoidally varying oscillatory flows created in an oscillatory U-shaped tube with the purpose of determining the force coefficients in a systematic manner as functions of t h e Keulegan-Carpenter number and t h e Reynolds number as well as the relative roughness of the cylinder.
T h e range of KC covered in the figure was rather small. I 11 3Asymptotic theory '2- 1- I 0. Data from X: Sarpkaya a , o, a: Bearman et al. Anatiirk T h e results of the asymptotic theory for the same Re number are also included in the figure. First consider the drag coefficient. As seen from the figure, there are three distinct regimes in the variation of Co with KC: Regarding the inertia coefficient, CM, from Fig.
T h e drag coefficient diagram includes also CQ versus Re variation for steady currents Fig. T h e figure is based on the results Chapter 4-' Forces on a cylinder in regular waves Sarpkaya Extended curves based on the following data KC: Steady current CD variation is reproduced from Fig.
Oscillatory flow data are from Sarpkaya a , Sarpkaya a and Justesen In-line force in oscillatory flow of the extensive study of Sarpkaya a and a and t h e study of Justesen It is apparent from t h e figure t h a t the drag coefficient varies with Re in the same m a n n e r as in steady currents. However, the drop in Cr with Re which is known as t h e drag crisis in steady currents, see Section 2.
For a given KC number, Cp first experiences a gradual drop with increasing Re number. Similar to the steady currents, this range of Re number may be interpreted as the lower transition regime see Section 2. Subsequently a range of Re number is reached where Cp remains approximately constant. This may be interpreted as the supercritical Re-number regime. Following t h a t , CD begins to increase with an increase in Re, interpreted as the upper transition. Re-number regime. Finally, the Cp coefficient reaches a plateau where it remains approximately constant with increasing Re.
This latter regime, on t h e other hand, may be interpreted as t h e transcritical. Regarding the inertia coefficient in Fig. W h e r e Cp experiences high values, CM experiences low ones. T h e increase in CM may be due to the weak vortex-shedding regime which takes place in the supercritical flow regime and particularly in t h e upper-transition flow regime.
Example 4. Effect of friction o n CD a n d CM In C h a p t e r 2, based on the experimental d a t a obtained for steady currents, it was demonstrated t h a t , for most of the practical cases, the friction drag is only a small fraction of the total drag Fig. Regarding t h e oscillatory flows, unfortunately no d a t a are available in t h e existing literature, therefore no conclusion can be drawn with regard to the effect of friction on the in-line force. Nevertheless, this effect may be assessed, utilizing Justesen's theoretical analysis.
T h e results depicted in Fig. Although the results are limited to small Re numbers, they nevertheless illustrate the influence of the friction on the force coefficients.
Regarding t h e drag coefficients, Fig. From numerical solution of Navier-Stokes equations in the subcritical Re number range.
Justesen , private communication , which is an extension of Justesen Asymptotic theory: Therefore, for large KC numbers, the drag portion of t h e in-line force may be considered to be due to pressure alone. Regarding the inertia coefficient, on the other hand, it is seen from Fig.
Sarpkaya and Isaacson Therefore it may be neglected in most of the practical cases. Clearly, the Morison representation is not extremely satisfactory with respect to t h e measured variation of the in-line force. T h e question how well the Morison equation represents the measured in-line force has been the subject of several investigations Sarpkaya and Isaacson, Clearly, t h e ability of the Morison equation to predict the force depends heavily on the KC number.
In the inertia-dominated region, S is rather small, therefore the Morison representation is rather good, b u t when the flow is separated, the Morison equation can not provide a complete description of the force variation.
T h e results have shown t h a t , in this way, a significant improvement has been obtained. Sarpkaya and Sarpkaya and Wilson Smooth cylinder. This lift force oscillates at a fundamental frequency different from the frequency of the oscillatory flow.
T h e time variation of the force is directly related to the vortex motions around the cylinder, as has already been discussed in Section 3. Obviously, if the flow around the cylinder is an unseparated flow very small KC numbers, Figs. T h e figure indicates t h a t , while the lift force first comes into existence when KC becomes 4 which is due to the asymmetry in the formation of the wake vortices; see Fig.
W h e n the analysis of the lift force is considered, the most important quantities are the fundamental lift frequency and the magnitude of t h e lift force. Regarding the fundamental lift frequency, this has been discussed in details in Sections 3. As regards the magnitude of the lift force, there are two approaches.
In one, the maximum value of t h e lift force is considered, while in the other t h e rootmean-square r. These may, in terms of the force coefficients, be written in the following forms: For the various flow regimes indicated in the figure, see Figs.
Lift force in oscillatory flow 'Lrms 1 - 10 15 KC 20 Figure 4. Experimental data from Justesen Figure 4. Willi amson Oscillatory flow data from Sarpkaya a. Steady-current Ci variation is reproduced from Fig. T h e figure indicates t h a t the lift force experiences two maxima, one at KC around 10 and a slight maximum at KC around This behaviour has been observed previously also by authors such as Maull and Milliner , Williamson , a n d Sarpkaya b, This obviously magnifies the aforementioned effect significantly.
Williamson's diagram is reproduced here in Fig. Williamson points out t h a t these peaks probably reflect an increase in the repeatability of the shedding patterns. Each peak corresponds to a certain p a t t e r n of shedding; namely, the first peak corresponds to t h e single-pair regime 7 Effect of roughness 15S numbers lie in the centre of t h e corresponding KC regimes, while the correlation is measured to be low and, as a result, C x r m s experiences minimum values at certain KC numbers because these KC numbers lie at the boundaries between the neighbouring KC regimes.
T h e figure includes also the steady current d a t a which are reproduced from Fig. As is seen, the effect of Re is quite dramatic see the discussion in Section 2. In addition to these effects, it increases t h e cylinder diameter, and the projected area.
Therefore it must be anticipated t h a t the effect of roughness upon the force coefficients can have some influence. T h e d a t a come from the work by Justesen It must be emphasized t h a t the experimental system in Justesen's work was maintained the same for all the three experiments indicated in the figure, and the experiments were performed under exactly the same flow conditions.
It is only the cylinder roughness which was changed. Therefore, the change in the force coefficients is directly related to the change in the roughness. Furthermore, it is clear t h a t CD increases with increasing roughness. Apparently CM is not influenced much with a further increase in the roughness. Regarding the decrease in CM, on the other hand, a clear explanation is difficult to offer.
T h e non-linear interaction between the vortex shedding and the hydrodynamic process generating t h e hydrodynamic mass - the mechanism behind the reduction in the hydrodynamic mass in the vortex-shedding-regime KC numbers - must occur more strongly in t h e case of rough cylinder, since the reduction in CM is much larger in this case t h a n in the case of smooth cylinder.
Sarpkaya a. Forces on a cylinder in regular waves is interesting to note that the way in which Co versus Re variation changes with respect to the roughness is quite similar to that observed in t h e case of steady currents Fig. As far as the lift force is concerned, Fig. Note that the depicted d a t a are from the same study as in Fig.