The article introduces a hybrid potential flow model (HPF) for predicting vortex-induced vibrations (VIV) around a circular cylinder. VIV is a significant phenomenon in various fields, such as submarine periscope resonance and power line galloping. The HPF model combines a source and doublet in uniform flow to represent the near-body problem and wake solution, respectively. It predicts forces due to vortex shedding and shedding frequencies for a circular cylinder in the subcritical Reynolds number regime. The model uses conformal mapping to extend its applicability to other bluff body geometries, making it a valuable tool for designing energy devices that exploit VIV. The HPF model incorporates experimental data to predict base pressure and separation points, providing a more accurate representation of the flow dynamics. The article also highlights the model's ability to estimate the oscillating lift force due to vortex shedding, a feature not commonly addressed by other theoretical models. The results obtained from the HPF model compare well with existing published literature, demonstrating its validity and potential for further development.
AI Generated
This summary of the content was generated with the help of AI.
Abstract
A Hybrid Potential Flow (HPF) model for flow around a circular cylinder in the subcritical Reynolds number range (\(300 \le Re \le 3\times 10^5\)) is developed using a combination of elementary flow solutions and empirical data. By joining this developed near-body solution with von Karman’s model for the vortex wake, a complete solution for flow around a circular cylinder is calculated. Results for oscillatory forces, including the transverse lift force, due to vortex shedding as well as shedding frequencies are then calculated and presented. With the complete solution for flow around a cylinder calculated, the HPF model can be used as a step to calculate the flow around other bluff bodies using conformal mapping, an approach that has been developed and presented by the authors in a related paper.
Notes
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
1 Introduction
It is a well-established and widely studied phenomenon that the flow of fluids around bluff bodies leads to the shedding of vortices in its wake. Asymmetric shedding of vortices can induce a fluctuating force transverse to the flow on the body, potentially causing self-induced oscillations. The resulting vortex-induced vibrations (VIV) are of significant interest in several fields—the resonance of periscope tubes on submarines, ‘galloping’ of power lines, ‘strumming’ of undersea cables, and vortex-induced oscillations of bridge decks, chimneys, and smokestacks [1].
In recent years, there has been renewed interest in the problem of flow around a bluff body. This is due to the potential to use large amplitude oscillations of VIV to generate energy [2, 3]. VIV can also form the basis for small-scale, low-cost energy devices. Optimal design of such devices will require a design tool to predict forces due to vortex shedding and shedding frequencies for different bluff body geometries.
Advertisement
In general, flow around a bluff body can be represented in two parts — the near-body problem and the vortex wake further downstream. The near-body problem involves representing flow separation on the surface of the bluff body and the separated boundary layers that continue downstream as free shear layers. Further downstream, the separated shear layers roll up into individual vortices in the wake, and the wake solution must represent this. For a complete theoretical solution, these two aspects of the bluff body solution must be combined [4]. However, the complex and unsteady dynamics of vortex formation in the region immediately downstream of the body has meant that theoretical treatment of flow around a bluff body has generally included some degree of empiricism.
There are several approaches to the theoretical modeling of this problem, such as free streamline models, vortex models, and vortex sheet models [5]. For free streamline models, Kirchhoff’s theory remains the foundation from which most begin. Here, the free shear layers separating from the bluff body are considered as surfaces of velocity discontinuity, enclosing a wake of constant pressure. The free shear layers, thus, form the boundary between the wake and outer potential flow. The challenge is to find the shape of these shear layers, so that the pressure distribution and, thus, the drag can be calculated. The pressure in the wake is assumed the same as freestream pressure, and the velocity along the free streamlines is considered constant and equal to freestream velocity. The magnitude of velocity at separation and all along the free shear layers are then the same as the freestream value. Transforming to the hodograph plane, the free streamline is a circle and allows easy determination of its shape in the physical plane.
The drag calculated using Kirchhoff’s method is far lower than experimentally observed values. A key reason for this result is the assumption that pressure in the wake is the same as freestream pressure. Experimentally, it is seen that this pressure, the base pressure, is much lower than the freestream value. Using a parameter k to relate base and freestream pressure and picking realistic values, Roshko [6] provides a semi-empirical solution for the near-body problem for a circular cylinder. Similar to Roshko, Woods [7] also employs hodograph methods and specifies magnitudes of velocity along the free streamlines, allowing conformal transformation of the complex velocity and complex potential planes.
Batchelor [8] proposed a limit solution model for flows as viscosity approaches zero. This model resulted in a closed wake, with the wake region now including a region of trapped vorticity, while external to this region flow is inviscid and irrotational. The model theoretically allowed calculation of the wake in terms of vorticity of the standing eddy and a constant, although Batchelor did not present any applications of the solution. This model has been further studied in detail, including in applications to a circular cylinder [9]. A drawback of Batchelor’s original model is that the resulting drag coefficient calculated for the body is zero.
Advertisement
Parkinson and Jandali [10] adopted a different approach to the hodograph methods described. For a circular cylinder, they noted that the free shear layers induced a reversed velocity along the bluff body surface beyond the separation point. The separation point appears as a confluence point where the retarded boundary layer and reversed flow meet. To simulate this, they placed two sources on the contour of the circular cylinder in some transform plane. The presence of sources leads to stagnation points on the body. Then, a conformal transformation is selected so that when mapping to a slit in the physical plane, the two stagnation points become separation points and the flow there is tangential as it leaves the body. In using two sources, the authors do not try to model mean flow in the wake. The main objective was to set separation velocities and model the effect of the wake on potential flow. A similar approach is used to model a normal flat plate. Bearman and Fackrell [11] built on the ideas of Parkinson and Jandali, developing a vortex lattice method for potential flow about bluff bodies of arbitrary shape, while still modeling the wake using surface sources.
Potential flow models do not generally account for the oscillatory forces due to periodic vortex shedding [12]. If the dimensions of and velocities in the wake are known, far-field control volume analyses such as the ones used by Chen [13] and later Sallet [14] can be used to calculate the lift force due to the vortex wake. In each of these methods, von Karman’s representation of an ideal vortex street [15] is used to depict the vortex wake. To calculate forces due to a vortex street behind a bluff body, theoretical models such as those of Chen [13] usually relied on wake velocity and one wake dimension that was measured experimentally.
Free streamline and vortex models require developing a solution for each bluff body geometry. Sallet’s method, while applicable to various bluff body geometries, is still dependent on knowledge of the wake behavior and dimensions of the wake. This is usually obtained experimentally. None of these theoretical methods allow quick estimation of vortex shedding behavior and forces due to vortex shedding by a variety of bluff body geometries.
In this paper, a new model for the near-body solution of flow around a unit circular cylinder, called the Hybrid Potential Flow (HPF) model, is presented. Joined with von Karman’s vortex wake solution, this model predicts forces due to vortex shedding as well as shedding frequency for a circular cylinder in the subcritical Reynolds number regime (\(300 \le Re \le 3\times 10^5\)). A motivation for developing a solution for the circular cylinder was to be able to extend it to a wide variety of other bluff body geometries through conformal mapping. Using a conformal mapping approach would remove the need for experimental data or developing a near-body solution for each bluff body. This conformal mapping approach is presented in a related paper [16] and in Matheswaran [17].
2 Methods
Consider the case of flow around a cylinder of unit radius, centered at the origin. The potential flow case for flow around such a cylinder is represented by a doublet of some strength \(\mu \) in uniform flow. In polar coordinates, the stream function for such a flow is
Here, \(v_\infty \) is the freestream velocity, \(\mu \) is the doublet strength, and \(r_1\) and \(\theta _1\) are the radial and transverse coordinate to any point in the flow. This is the general case of potential flow around a circular cylinder, and does not model separation in the wake. Stagnation points are at (−1,0) and (1,0), marked as A and B in Fig. 1.
Fig. 1
Potential flow around a circular cylinder of unit radius. Forward and rear stagnation points are labeled as A and B, respectively. Flow is from left to right
The HPF model starts with this same formulation, but now a source of some unknown strength Q is placed downstream of the doublet to model the separated wake region. The stream function for such a flow is
Here, \(\theta _2\) is the transverse coordinate from the location of the source. To still represent flow around a circular cylinder of unit radius centered at the origin, the forward stagnation point A must continue to be at (-1,0). The stagnation streamline will now no longer pass through the rear stagnation point B (1,0). For the forward stagnation point A, \(\theta _1 = \theta _2 = \pi \) and \(r_1\) = 1. Using these values in (3),
The challenge now is to pick a combination of Q and a such that for a given Reynolds number in the subcritical range (\(300 \le Re \le 3\times 10^5\)), the stagnation streamline
1.
represents the surface of the cylinder upstream of separation.
2.
represents the free shear layers in the wake downstream of the separation point.
In addition, the right combination of Q and a is that which will result in the location of the separation point and the pressure at separation similar to what is observed experimentally. The presence of the source will now necessitate that \(a<\) 1. The Reynolds number is defined as follows:
$$\begin{aligned} Re = \frac{v_\infty D}{\nu }, \end{aligned}$$
(5)
where \(\nu \) is the kinematic viscosity of the fluid, and D the characteristic length (in the case of a cylinder, the diameter).
This approach is illustrated in Fig. 2. A doublet of radius a, centered at origin, and a source of strength Q placed at (1,0) is used to model flow around a circular cylinder of unit radius. The stagnation streamline represents the surface of the cylinder prior to separation and after separation represents the free shear layers in the wake. The only flow elements used are the doublet at origin and the source at (1,0) — the contour of the circular cylinder is simply provided for reference. Radial coordinates \(r_1\), \(r_2\) are marked, and transverse coordinates \(\theta _1\), \(\theta _2\) are measured counter-clockwise from the x-axis.
Fig. 2
Combination of source and doublet in uniform flow. The doublet is centered at the origin, its extent represented by the dashed line
The right combination of Q, a is solved for by imposing appropriate boundary conditions. A complex potential approach is convenient when applying boundary conditions. The complex potential and complex conjugate velocity for this flow are
$$\begin{aligned} \frac{dw}{dz}= & {} u - iv = v_\infty \left( 1 - \frac{a^2}{z^2}\right) + \frac{Q}{2\pi (z-z_s)}. \end{aligned}$$
(7)
Here, the doublet is centered at origin, \((0+0i)\), and location of the source is \(z_s = (1 + 0i)\).
Two boundary conditions are imposed to solve for a unique combination of Q and a.
1.
Zero flow velocity at the forward stagnation point: For a circular cylinder of radius one unit centered at origin, \(z = -1+0i\) is a stagnation point.
2.
Velocity at separation (\(v_s\)): Experimentally, it is seen that the velocity at separation (\(v_s\)) is greater than the freestream velocity, defined as \(v_s = k v_\infty \), where k is the base pressure parameter.
Enforcing these boundary conditions requires knowledge of the base pressure as well as the location of separation point, both of which vary with Reynolds number. Determining the base pressure and separation point as a function of Reynolds number is described in the following section.
2.1 Base pressure & separation point
The time-averaged pressure over the base of the cylinder (portion of the cylinder exposed to the wake) is nearly constant [4, 6, 10], denoted as the base pressure \(p_b\). The customary expression for pressure coefficient[18] is
Here, p and \(p_\infty \) are pressures at a point and at freestream conditions respectively, \(\rho \) is the density, and \(v_\infty \) is again the freestream velocity. This can be simplified further. From Bernoulli’s equation, pressure at any point is
$$\begin{aligned} p_o = p + \frac{1}{2}\rho v^2, \end{aligned}$$
At the separation point, p becomes the base pressure \(p_b\), and v is the separation velocity \(v_s\). Defining base pressure parameter as \(k = \frac{v_s}{v_\infty }\), Eq. (13) now becomes:
The base pressure coefficient \(C_{p_b}\) is, thus, defined in terms of the base pressure parameter.
Prior theoretical models have taken different approaches to determining k for various Reynolds numbers. In Kirchhoff’s free streamline theory, it is assumed that pressure in the separated wake is the same as freestream pressure, leading to k = 1. Subsequent free streamline models [6, 10] used more realistic values of k, usually determining it experimentally. For a circular cylinder, assuming the curvature of the streamline is the same as that of the cylinder, Roshko’s method [6] calculated a unique value of k for every value of time-averaged separation angle (\(\overline{\beta _s}\)). However, there is a marked difference between the values of k observed experimentally for a given \(\overline{\beta }_s\) and those predicted by Roshko’s relation. Roshko attributes this to the assumption about the streamlines’ curvature.
In the present HPF model, to capture the variation of k with Reynolds number, an exponential model is developed. A compilation of experimental measurements of the variation of base pressure coefficient \(C_{p_b}\) with Reynolds number in the subcritical regime is shown in Fig. 3. This experimental data has been collected from several sources [19‐24], with turbulence intensities in each of the studies not being more than 0.4%. Zdravkovich [5] and Norberg [25] note that experimental observations include a large degree of scatter due to the effect of turbulence intensity. The data used here has not been corrected for turbulence intensity. The data used are of varying blockage ratios, with Gu et al. [22] noting a blockage ratio of 6%. No corrections are applied for blockage effects. From this set of data, an equation for k as a function of Reynolds number can be developed:
$$\begin{aligned} k = 1.51 - 0.211{\text {e}}^{(-0.000121 Re)}. \end{aligned}$$
(15)
The value of k and thus \(C_{p_b}\) can now be predicted as a function of Reynolds number.
Fig. 3
Variation of base pressure coefficient (\(C_{p_b}\)) with Reynolds number. Data from [19‐24]. Also shown are the base pressure coefficient as predicted by the exponential \(k-Re\) model, given by Eq. (15)
For bluff bodies such as a flat plate (normal to the flow) or a wedge, sharp corners can reliably be assumed as separation points. This is not possible for a circular cylinder. The separation points on either side are not fixed and oscillate with eddy shedding. The range of variation of separation points is less than a degree at very low Reynolds numbers (Re =50), but can quickly rise to over \(5^{\circ }\) at Re = 160 [26]. With increase in Reynolds number, this range increases. Maekawa and Mizuno [27] observed the separation point oscillating between \(78^{\circ }\) and \(90^{\circ }\) for \(3.7 \times 10^4 \le Re \le 6.5 \times 10^4\), although cylinder span [28] and the state of the boundary layer (laminar or turbulent) over the cylinder have an effect on this range. For the ideal case of 2D steady flow past a smooth circular cylinder, flow is only dependent on one parameter, the Reynolds number. For various flow regimes, the relationship between time-averaged separation point (\(\overline{\beta }_s\)) and Reynolds number can be identified. As proposed by Jiang [28], the time-averaged separation point in the subcritical regime can be written as a function of Reynold’s number:
Since flow is symmetric about the center line of the wake, \(\overline{\beta }_s\) is identical on both sides of the circular cylinder. Jiang’s results [28] correspond well to experimental observations and are used to predict the time-averaged separation angle (\(\overline{\beta }_s\)) with varying Reynolds number in this study.
2.2 Enforcing boundary conditions
For the subcritical Reynolds number range, the time-averaged separation point \(\overline{\beta }_s\) and base pressure parameter k can now be written as a function of Reynolds number (Eqs. (15) and (16), respectively). Finding the combination of source strength and doublet radius is a matter of applying appropriate boundary conditions as previously described so that upstream of the separation point, the stagnation streamline represents flow around a circular cylinder, while downstream, it represents the separated boundary layers as shear layers. Application of boundary conditions is discussed here.
For a circular cylinder of unit radius centered at origin with uniform flow directed along the positive real axis, \(z = -1+0i\) is a stagnation point. Here,
Velocity at any point in the flow field is given by Eq. (7). Knowing that the magnitude of velocity at separation is \(v_s = kv_\infty \), and the second boundary condition can be stated as follows:
This second boundary condition must be applied at the point of separation on the surface of the cylinder, \(z = 1{\text {e}}^{i\beta _s}\), with \(\overline{\beta }_s\) given by Eq. (16). The source is located at \(z_s = 1 + 0i\) as previously mentioned. Substituting for \(\frac{dw}{dz}\) using Eq. (7), and for \(a^2\) using Eq. (19) reduces Eq. (20) to a quadratic equation in Q. The unknown source strength can now be determined, followed by the doublet radius. This series of steps and the resulting quadratic equation is described in Appendix 1 as Eq. (41). The resulting equation leads to two values of Q. Of this, only the value that leads to real, positive values of a is considered.
Thus, for given values of Reynolds number, \(\overline{\beta }_s\), and k, the required combination of source and doublet strengths to approximate flow around a circular cylinder of unit radius and its wake can be calculated.
2.3 Wake structure and forces
The combination of source and doublet in uniform flow described thus far is used to represent the near-body solution around a circular cylinder of radius one unit. The velocity and pressure distribution along the front of the cylinder (i.e., the stagnation streamline) can be determined using Eq. (7). Once the value of base pressure parameter k is calculated for a given Reynolds number, the base pressure coefficient is known using Eq. (14). The pressure in the wake region is assumed constant, as seen experimentally. Immediately downstream of the cylinder, the stagnation streamline represents the separated free shear layers. The velocity in the free shear layers right after separation is set as constant and equal to the separation velocity, \(v_s = kv_\infty \).
Further downstream, it is seen experimentally that the shear layers roll up into alternating vortices. Here, this is represented by von Karman’s ideal vortex street wake [15]. Any complete solution must join the near-body solution with the vortex street wake. As noted by Roshko [4], one way to do this is to equate the drag: the drag from the near-body model developed must be the same as that from application of the momentum theory to von Karman’s representation of the vortex street [15, 29]. To represent this vortex wake, one must have an idea of the velocities in the wake and its dimensions.
Results from the developed near-body model provide these values. The width of the wake, h, is provided by the spacing between the free shear layers in the near-body solution. The lateral spacing of vortices in the wake, l, is given by one of von Karman’s stability parameters for a vortex wake, \(\frac{h}{l} = 0.281\) [15]. It is still necessary to determine velocities and circulation in the wake for a complete solution.
2.3.1 Velocity and circulation in the wake
Not all of the vorticity shed from a cylinder ends up in the individual rolled up vortices of the wake. The vorticity shed from each side of the bluff body can be considered as a vortex band (the free shear layer). If \(v_1\) and \(v_2\) are the time-averaged velocities on the edges of the band (or shear layer) such that \(v_1\) is the outer band and \(v_2\) is the inner band, then the rate at which circulation moves downstream is [30]:
To accurately evaluate the amount of vorticity shed from a separation point during a shedding cycle, Eq. (21) should be integrated over the shedding period [31]. An approximation can be made by setting the velocity of the inner shear layer as zero since \(v_2\) is an order of magnitude larger than \(v_1\), following the experimental observations of Fage and Johansen [32], as well as Roshko’s approach [4]. Then, Eq. (21) becomes
where \(v_s = kv_{\infty }\) is the separation velocity as previously described. This rate at which vorticity is shed must be equal to rate of transport of circulation downstream in the vortex street, \(n\Gamma \). Here, n is the shedding frequency and \(\Gamma \) is the circulation associated with each individual vortex. Fage and Johansen [32] note that only 40–60% of the vorticity shed from the separation point ends up in the vortex street, reasoning that vorticity from one side moves into the rolling up vortices of the other side and cancels out. Using \(\epsilon \) as the portion of shed vorticity that ends up in the vortex street, and rewriting the left side of Eq. (21) as \(n\Gamma \),
\(\epsilon \) is an unknown that must still be determined.
If v is the velocity of vortices in the vortex street relative to the freestream and l is the lateral spacing of vortices in the vortex street as mentioned, n can be rewritten, with Eq. (23) becoming
Following Roshko’s approach, \(\Gamma \) can be removed from the equation by using one of von Karman’s stability parameters for a vortex street, \(\Gamma /vl = 2\sqrt{2}\). Then,
Eq. (26) relates the velocity in the wake with the freestream velocity.
2.3.2 Drag and lift due to vortex wake
To join the near-body and vortex street solutions, the drag predicted by the two must be equated. Setting up a control surface around the cylinder and applying the momentum equation in the direction of fluid flow, von Karman [15] writes the drag coefficient due to a vortex street as
Here, the values of h and d are known. Since pressure drag due to the near-body solution must be the same as that due to the vortex street, the value of \(C_d\) in the above equation is set as that found using the near-body solution. This allows finding \(\epsilon \) and, working backwards, the value of \(\Gamma \).
With the complete solution now in place, the oscillating lift force due to vortex shedding can now be calculated using a momentum approach, as per Chen [13]:
$$\begin{aligned} L' = \rho v\Gamma . \end{aligned}$$
(30)
This expression is different from the steady lift due to circulation for a body in parallel flow, \(L = \rho v_\infty \Gamma \), being instead dependent on v, the velocity of vortices in the vortex street relative to the freestream.
With the velocities and circulations in the wake known, shedding frequency and Strouhal number can be calculated using Eq. (23).
3 Implementation
For a circular cylinder of unit radius in uniform flow at subcritical Reynolds numbers, a step-by-step implementation of the HPF model is described here.
1.
At the Reynolds number of interest, the time-averaged separation point \(\overline{\beta }_s\) and base pressure parameter k are calculated using Eqs. (16) and (15).
2.
The right combination of Q and a are now calculated using Eq. (41). It is now possible to calculate velocity and pressure distribution around the cylinder. From the pressure distribution, the drag coefficient as predicted by the near-body solution is also calculated.
3.
The near-body solution has now been constructed. Plotting the stagnation streamline, which represents the separated free shear layers immediately downstream, the width of the wake h can be determined.
4.
Downstream, the shear layers roll up into alternating vortices. The drag coefficient calculated in step 2 is used in the vortex street solution, Eq. (29).
5.
The near-body and vortex street solution have now been joined. \(\epsilon \) can be solved for, following which it can be used to determine velocities in the wake using Eq. (26). Lastly, \(\Gamma \) can be calculated using the stability parameter, \(\frac{\Gamma }{vl} = 2\sqrt{2}\).
6.
Once \(\Gamma \) is known, oscillating lift force is calculated using Eq. (30).
4 Results and discussion
4.1 Cylinder wake
Using the HPF model, the near-body solution for the time-averaged wake behind a circular cylinder is calculated at a given Reynolds number. This, joined with von Karman’s solution for a vortex street, is used to calculate forces induced due to vortex shedding.
The near-body solution for flow around a unit circular cylinder at \(Re = 1.7 \times 10^4\) as predicted by the Hybrid Potential Flow model is shown in Fig. 4. For this Reynolds number, the mean separation point (\(\overline{\beta }_s\)) and base pressure parameter (k) are first calculated, followed by the unique combination of source and doublet strength corresponding to these two cases. These are listed in Table 1. As seen in Fig. 4, the stagnation streamline mirrors the surface of the cylinder. After separation, the stagnation streamline continues downstream and represents the time-averaged position of the separated shear layers. It is the boundary between the outer potential flow and the wake region. The spacing between the shear layers represents the wake width, h. As mentioned previously, the only flow elements used are a source and doublet. The outline of the unit cylinder is provided for reference only. Also shown in Table 1 are parameters for the near-body solution at a higher Reynolds number. These results are presented for a dimensioned cylinder of radius 1 ft, leading to dimensioned values of all other quantities.
Table 1
Parameters in HPF model for flow around a unit circular cylinder at two different Reynolds numbers
Re
Q(\(ft^2/s\))
a (ft)
\(\overline{\beta _s}(^\circ \))
k
1.7 \(\times 10^4\)
4.47
0.849
82.67
1.48
6 \(\times 10^4\)
14.83
0.859
80.86
1.51
Here, \(\overline{\beta }_s\) is shown measured clockwise from the forward stagnation point, calculated using Eq. (16)
Fig. 4
Flow around a unit circular cylinder at \(Re = 1.7 \times 10^4\)
Fig. 5
Pressure distribution along the stagnation streamline (which represents the surface of the cylinder). Here, Re = \(1.7 \times 10^4\). Pressure in the wake region is set as the base pressure parameter, \(C_{p_b}\)
4.2 Forces on cylinder
Since the velocity along the surface of the cylinder prior to and at separation, as well as the pressure in the base region is known, the pressure distribution around the cylinder and, thus, the drag can be calculated. The pressure coefficient around the top half of the cylinder at a Reynolds number of \(1.7 \times 10^5\) is shown in Fig. 5 calculated using this model. The variation in drag coefficient with Reynolds number in the subcritical range predicted by the HPF model is shown in Fig. 6. Experimental results from Hoerner [33] and Chen [13] are also shown.
Fig. 6
Variation of drag coefficient with Reynolds number. Results from the HPF model are compared to experimental results from [13, 33]
Further downstream, the free shear layers roll up into alternating vortices. The near-body solution and von Karman’s solution for the vortex wake are joined by equating drag. Wake dimensions, fraction of vorticity \(\epsilon \) from the shear layers that ends up as individual vortices in the wake, strength of individual vortices \(\Gamma \) in the wake, and the rms lift coefficient per span for two Reynolds numbers are presented in Table 2. The variation of \(C_{l_{rms}}\) with Reynolds number predicted by the HPF model is shown in Fig. 7, and compared to experimental measurements from West and Apelt [34], and Norberg [25]. The maximum fluctuating lift coefficient predicted by the HPF model is compared to theoretical estimates by Chen [13] as well as experimental measurements by Gerrard [35] and Norberg [19] in Fig. 8.
Table 2
Wake characteristics and RMS lift coefficient for a unit circular cylinder
Re
h (ft)
l (ft)
\(\Gamma (ft^2/s)\)
\(\epsilon \)
\(C_{l_{rms}}'\)
\(1.7 \times 10^4\)
3.15
11.22
5.09
0.32
0.35
\(6 \times 10^4\)
2.98
10.60
18.86
0.33
0.41
Fig. 7
Variation of rms lift coefficient with Reynolds number for a circular cylinder. Results from the HPF model are compared to prior published experimental results from [25, 34]
Fig. 8
Variation of maximum fluctuating lift coefficient with Reynolds number for a circular cylinder. Results from the HPF model are compared to prior published theoretical and experimental results from [13, 19, 35]. The higher turbulence intensity in [19] has been noted
As is seen in Fig. 6, the HPF model predictions for drag compare well with experimental observations. Figures 7, 8 show that predicted lift values in the subcritical range compare reasonably well too, although they are consistently lower than experimental measurements. As noted by Zdravkovich [5] and Norberg [25], there is a large amount of scatter in experimental observations of fluctuating lift coefficient due to the dependence of the unsteady process of vortex shedding on free stream turbulence intensity, surface finish, model aspect ratio, and end effects. The fluctuating lift is also highly dependent on the degree of three-dimensionality in the wake of the cylinder. Further discrepancies in lift and drag values arise due to the assumption that the wake dimensions predicted using von Karman’s ideal spacing are constant throughout the wake. While the ratio of lateral and longitudinal spacing between vortices in the wake remain constant, it is well understood that the wake behind a bluff body will begin to expand as it moves downstream. Despite these approximations, the results presented above indicate that the HPF model is a valuable tool in predicting wake forces and dimensions behind a circular cylinder, and thus, apt to use in a conformal mapping approach extended to other bluff body geometries. This approach has been described in a related paper [16].
4.3 Strouhal number and shedding frequency
The Strouhal number (St) is a non-dimensional quantity relating predominant shedding frequency n, fluid velocity \(v_\infty \), and a reference length D of the bluff body.
$$\begin{aligned} St = \frac{nD}{v_\infty }. \end{aligned}$$
(31)
Strouhal numbers and shedding frequencies predicted by the HPF are compared to the well-known St-Re curve for a circular cylinder. In Table 3, Strouhal numbers predicted by the HPF at four Reynolds numbers in the subcritical range is compared to a compilation of published data from Lienhard [36], reproduced in Blevins [1]. These Reynolds numbers were picked as they correspond to experimental testing conditions, presented in [17]. The reference length used in all St calculations is the cylinder diameter.
The data presented in Blevins and Lienhard has an uncertainty of \(\pm 5\%\) [36]. The values predicted by the HPF model are lower than the experimentally observed values. They do, however, remain relatively constant between \(10^3 \le Re \le 10^5\), as is the case with experimental measurements. It can be reasoned that the lower Strouhal number predicted is due to the approximations made with estimating the vorticity shed from the bluff body and the amount that ends up in the vortex street (Eq. 21). Lower estimates of \(\epsilon \) will lead to lower shedding frequencies and thus Strouhal number.
Table 3
Comparison of predicted and observed Strouhal numbers for cylinder
The comparisons between the predicted values and previously published values of St indicate the validity of Strouhal numbers and shedding frequencies predicted by the HPF model, as well as the ability of the HPF model to estimate vortex shedding behavior for a circular cylinder.
5 Conclusions
The Hybrid Potential Flow model discussed in this paper allows the prediction of flow around a stationary circular cylinder in the subcritical regime. It constructs a new and complete solution using a combination of a source and doublet in uniform flow and experimental data, combined with an existing theoretical solution for the vortex wake. This is an approach that has not been previously used in literature. Using the near-body solution, wake dimensions, strength of vortices in the wake, and forces due to vortex shedding can be predicted. The results predicted compare well with existing published literature. Unlike most prior theoretical models that only calculate mean drag on the cylinder, this method estimates the oscillating lift force due to vortex shedding. By compiling experimental data from several sources, a reliable model to predict base pressure parameter k as a function of Reynolds number has been constructed.
As previously mentioned, a significant motivation in developing the HPF model is to have a reliable solution for flow around a circular cylinder that can then be used in a conformal mapping approach. This foundation will allow quick prediction of the forces due to vortex shedding and shedding frequencies for different geometries, and serve as a design tool for energy devices that use VIV as its basis. This conformal mapping approach is described in Matheswaran et al. [16].
There are avenues for further improvement of the HPF model. The first is an improved k-Re model. The k-Re model developed here uses experimental data from a variety of sources. These sources were selected for the experimental conditions used—low turbulence intensity and surface roughness. An improved k-Re model that accounts for different turbulence intensities as well as surface roughness of the model will lead to an improved k-Re model, and greater applicability of the HPF solution.
The second is to include the effect of VIV on predicted forces and shedding frequencies. The HPF model and conformal mapping predict the vortex shedding behavior for rigid stationary bluff bodies of various geometries. For a body that is elastic or not stationary, large amplitude VIV can affect vortex patterns in the wake, which in turn affects body motion. A complete model could look to include bluff body material properties (mass, stiffness and damping) as well as amplitudes of vibration when predicting vortex shedding behavior.
Declarations
Competing interests
The authors declare no competing interests.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Velocity at any point in the flow field is given by Eq. (7). The separation point is at \(z = {\text {e}}^{i\beta _s}\) and the source at \(z_s = 1 + 0i\). Then, at the separation point, Eq. (7) becomes
$$\begin{aligned} \frac{dw}{dz} = u - iv = v_\infty \left( 1 - \frac{a^2}{{\text {e}}^{2i\beta _s}}\right) + \frac{Q}{2\pi ({\text {e}}^{i\beta _s}-1)}. \end{aligned}$$
As previously mentioned, \(z_1, z_2, z_3, z_4\) are all known constants, and \(v_\infty \) is known. The value for the base pressure parameter, k, is also known from Eq. (15). Two roots (values of Q) are, thus, obtained. Of this, only the value that leads to real, positive values of a is considered.
