Vortex shedding from bluff bodies: a conformal mapping approach

Abstract

A model to calculate flow around bluff bodies of various geometries is presented. A conformal map between the plane of the bluff body and the plane of a unit circular cylinder is established by using a combination of Karman–Trefftz transformations and Fornberg’s method. Flow in the circle plane is calculated using the authors’ Hybrid Potential Flow (HPF) model and mapped back to the shape plane. By joining this calculated near-body flow with von Karman’s model for a vortex wake, forces due to vortex shedding and shedding frequencies are calculated. In this manner, a complete solution for the flow around bluff bodies of various geometries is established. Results for two shapes are presented, along with recommendations for further work.

1 Introduction

The problem of flow around a bluff body, which leads to separation and the formation of a broad wake, is a fundamental one in fluid mechanics. Asymmetric shedding of vortices in the wake of a bluff body can induce fluctuating forces on the body that are both transverse and in-line with the flow. This can result in self-induced oscillations, or vortex-induced vibrations (VIV) of the body. The complexity of wake dynamics and the formation of a vortex street has meant that theoretical models have usually included some level of empiricism.

Several studies have, in recent years, focused again on the problem of flow around a bluff body. A reason for this is the potential to extract energy from the large-amplitude oscillations of a body undergoing VIV [1, 2]. VIV can also form the basis for low-cost energy devices. Effective design of such devices will necessitate predicting shedding frequencies and forces due to vortex shedding. While circular cylinders have been the most commonly studied geometry in literature, bluff bodies of various cross-sections shed vortices [3].

Theoretical representations of flow around a bluff body usually use free streamline models, vortex models, and vortex sheet models. Free streamline models generally use Kirchhoff’s theory as the starting point [4,5,6], denoting the free shear layers as surfaces of velocity discontinuity and thus enclosing a wake of constant pressure. The free shear layers are thus the boundary between the wake and potential flow exterior to it. By considering velocity as constant along it, the free streamline transforms to a circle in the hodograph plane and its shape can thus be calculated in the physical plane.

Conformal mapping has long been a tool in fluid mechanics, and conformal maps between a circle and an airfoil have been used to calculate the flow around an airfoil. For bluff bodies, the wake source model developed by Parkinson and Jandali [7] used sources placed on the contour of a cylinder in some transform plane. The sources lead to stagnation points on the body. A conformal transformation must then be selected so that when mapping to a slit in the physical plane, the two stagnation points become separation points. A similar approach can be applied to a flat plate. This method can be applied to other bluff bodies, but a suitable conformal transformation must be selected each time. Bearman and Fackrell [8] further developed the ideas of Parkinson and Jandali, developing a vortex lattice method for arbitrary bluff bodies, while still modeling the wake using surface sources. Both of these approaches, while being able to solve for the flow around different bluff bodies, did not provide information on shedding frequencies and forces due to vortex shedding.

That shortcoming is addressed in the methods presented here. In a related paper, Matheswaran and Miller [9] presented the Hybrid Potential Flow (HPF) model for flow around a unit circular cylinder using a combination of elementary flow solutions and experimental data. The flow field, wake velocities, and dimensions can be calculated using the near-body solution developed. By joining this near-body solution with von Karman’s representation of the vortex wake [10], the HPF model allows prediction of shedding frequencies and forces due to vortex shedding on the cylinder anywhere in the sub-critical Reynolds number range. The results from the HPF model compare well to experimental observations, showing its validity.

In this paper, the authors extend this approach to bluff bodies of various cross-sections using a conformal mapping approach. Following previous work by DeLillo [11, 12], a series of Karman–Trefftz transformations and Fornberg’s method are used. The domain exterior to an arbitrary geometry is mapped to the domain exterior to a unit circular cylinder. Flow is solved in the circle plane using the near-body solution in the HPF model, and then mapped back to the plane of the arbitrary geometry. In this manner, flow around a bluff body of arbitrary cross-section can be solved. By joining this with von Karman’s solution for the vortex wake, the resulting shedding frequencies and forces due to vortex shedding can be calculated.

In the subsequent sections, the conformal mapping techniques are first discussed, followed by application of the HPF model and conformal mapping to three bluff body shapes. Obtained results are then presented and compared to previously published experimental values. Finally, the class of bluff bodies to which this technique is applicable is discussed, followed by recommendations for future work. In the interest of brevity, the HPF model is not discussed in this paper. For this, the reader is referred to Matheswaran [13] and the related paper by Matheswaran and Miller [9].

2 Methods

A brief overview of the two conformal mapping techniques used is provided here. Since Fornberg’s method maps the domain exterior to a unit disk to the domain exterior to a smooth closed curve, it is necessary to first transform the geometry in question first to a smooth closed curve. This is done through a series of Karman–Trefftz transformations. Fornberg’s method can then be used to calculate a conformal map between the planes of the smooth closed curve and the unit circle.

2.1 Karman–Trefftz transformations

Consider an arbitrary geometry in the physical ($\mathcal {Z}$) plane, such as that shown in the first panel of Fig. 1. To convert this geometry to a smooth, closed curve, each corner will need to be smoothed. Let the corner to be smoothed be denoted by $Z_1$ and an inner point to the geometry be $Z_2$. Then, the classical Karman–Trefftz transformation from the $\mathcal {Z}$-plane to the $\zeta$-plane is as shown below:

$$\begin{aligned} \frac{\zeta - \zeta _1}{\zeta - \zeta _2} = \left( \frac{Z- Z_1}{Z- Z_2}\right) ^{1/\beta }. \end{aligned}$$

(1)

Here, $Z_1$ maps to $\zeta _1$ and $Z_2$ maps to $\zeta _2$. $\beta$ is calculated as

$$\begin{aligned} \beta = \frac{2\pi - \lambda }{\pi }, \end{aligned}$$

(2)

where $\lambda$ is the interior angle of the corner to be smoothed. Then

$$\begin{aligned} k(Z) = \zeta = \left( \zeta _1 - \zeta _2\left( \frac{Z - Z_1}{Z - Z_2}\right) ^{1/\beta }\right) /\left( 1 - \left( \frac{Z-Z_1}{Z-Z_2}\right) ^{1/\beta }\right) . \end{aligned}$$

(3)

For a geometry with multiple m corners to be smoothed, successive Karman–Trefftz maps must be used. The Karman–Trefftz map can then be written as a composition:

$$\begin{aligned} k(Z) = \zeta = k_m\circ k_{m-1} \circ ... k_1 (Z). \end{aligned}$$

(4)

Karman Trefftz maps can be explicitly inverted. Thus, for a single transformation, the inverse can be written as

$$\begin{aligned} k^{-1}(\zeta ) = Z = \left( Z_1 - Z_2\left( \frac{\zeta - \zeta _1}{\zeta - \zeta _2}\right) ^{\beta }\right) /\left( 1 - \left( \frac{\zeta - \zeta _1}{\zeta - \zeta _2}\right) ^{\beta }\right) . \end{aligned}$$

(5)

For multiple Karman–Trefftz transformations, the inverse is again written as a composition:

$$\begin{aligned} k^{-1}(\zeta ) = Z = k_1^{-1}\circ k_2^{-1} \circ ... k_m^{-1}(\zeta ). \end{aligned}$$

(6)

Shown in Fig. 2 is the process of applying multiple Karman Trefftz maps to a geometry in the $\mathcal {Z}$-plane to result in a smooth closed curve in the $\zeta$-plane. Although the smooth domain is not always symmetric in the $\zeta$-plane, since it is an intermediate step, it is not a cause for concern.

Fig. 1

Conformal mapping approach using Karman–Trefftz and Fornberg maps. A point in the circle plane z can be calculated in the physical plane $\mathcal {Z}$ as $\mathcal {Z} = k^{-1}(h(z))$

Fig. 2

Karman–Trefftz transformation for a wedge. Here, $m = 3$

2.2 Fornberg’s method

Fornberg’s method conformally maps the domain exterior to a unit disk to the domain exterior to a smooth closed curve. As noted by DeLillo and Sahraei [12], applying Fornberg’s method requires that the smooth closed curve resulting from the Karman–Trefftz transformations first be parameterized as a smooth function with continuously turning tangent (no corners) [12]. This is done by fitting the x, y coordinates of the smooth closed curve with two periodic cubic splines parametrized by S. Then, the form of the Fornberg map that maps the exterior of a unit disk $\partial \Omega$ to the exterior of a smooth closed curve $\gamma (S)$ is

$$\begin{aligned} h(z) = a_1 z + a_0 + \sum _{k=1}^{\infty } a_{-k}z^{-k}. \end{aligned}$$

(7)

A Newton-like method is used to find the boundary correspondence $S = S(\theta )$, so that

$$\begin{aligned} h(e^{i\theta }) = \gamma (S(\theta )). \end{aligned}$$

(8)

Implementation of the periodic cubic splines and Fornberg map is done following the work of DeLillo and Sahraei [12]. For computational purposes, the infinite series is truncated to allow calculation of 256 Fourier points.

2.3 Conformal map from the physical plane ($\mathcal {Z}$) to the circle plane (z)

With the Karman–Trefftz transformations and Fornberg map calculated, the conformal map from the physical $\mathcal {Z}$-plane to the circle z-plane can be written as a composition of the two. Since h is the Fornberg map from the circle plane z to the plane of the smooth shape $\zeta$, and $k_1, k_2...k_m$ are the successive Karman Trefftz maps from the physical plane $\mathcal {Z}$ to the $\zeta$-plane, the conformal map f from the z-plane to the $\mathcal {Z}$-plane is written as

$$\begin{aligned} \mathcal {Z} = f(z) = k^{-1}\circ h = k_{1}^{-1}\circ k_{2}^{-1} \circ ... k_m^{-1} \circ h(z), \end{aligned}$$

(9)

that is

$$\begin{aligned} \mathcal {Z} = f(z) = k^{-1} \circ h(z). \end{aligned}$$

(10)

This is illustrated in Fig. 1.

2.3.1 Velocity calculations

Velocity at every point in the physical $\mathcal {Z}$-plane can be found by calculating the derivative of the conformal map. Let w(z) be the complex velocity potential in the z-plane. Keeping in mind that the conformal map is written as $\mathcal {Z} = f(z)$, complex velocity potential W in the $\mathcal {Z}$-plane can be written as

$$\begin{aligned} W(\mathcal {Z}) = W(f(z)) = w(z) = w(f^{-1}(\mathcal {Z})). \end{aligned}$$

(11)

Then, complex conjugate velocity $\overline{V}$ in the $\mathcal {Z}$-plane is

$$\begin{aligned} \overline{V} = \frac{dW}{d\mathcal {Z}} = \frac{d}{d\mathcal {Z}}w(f^{-1}(\mathcal {Z})) = w'(z)\frac{dz}{d\mathcal {Z}} = \frac{w'(z)}{f'(z)}. \end{aligned}$$

(12)

Equation (12) can be modified to calculate the freestream velocity $V_\mathcal {Z}$ in the $\mathcal {Z}$-plane. Writing $v_\infty$ as the freestream velocity in the z-plane and using $z = \infty$, the freestream velocity in the $\mathcal {Z}$-plane is

$$\begin{aligned} \overline{V}_\mathcal {Z} = \frac{w'(\infty )}{f'(\infty )}=\frac{\overline{v}_\infty }{f'(\infty )}. \end{aligned}$$

(13)

Here, $\bar{v}_\infty$ indicates it is the complex conjugate of the freestream velocity in the z-plane. Thus, velocity at every point in the $\mathcal {Z}$-plane can thus be calculated with Eqs. (12), (13) once the derivatives of the conformal map $f'(z)$ and $f'(\infty )$ have been computed.

2.3.2 Joining the near-body solution and the vortex wake

The conformal mapping approach discussed so far allows calculation of the near-body solution. Further downstream, it is well known that the separated shear layers roll up into individual vortices. Joining the near-body solution with the vortex wake is done in the same manner as discussed in the HPF model, and is recounted here.

With the conformal map, flow everywhere around the bluff body and on its surface can be calculated. Pressure in the region aft of the separation points, the base pressure, is considered constant like in the HPF model [9]. This base pressure is given by

$$\begin{aligned} C_{p_b} = 1 - \Big (\frac{V_s}{V_\mathcal {Z}}\Big )^2, \end{aligned}$$

(14)

where $C_{p_b}$ is the base pressure coefficient, $V_s$ is the velocity at separation, and $V_\mathcal {Z}$ is the freestream velocity, both in the $\mathcal {Z}$-plane. Knowing the pressure distribution around the body allows calculation of the pressure drag.

Downstream of the bluff body, the free shear layers roll up into individual vortices. To join the near-body solution with the solution for the von Karman vortex wake [10], the drag is equated. That is, the pressure drag calculated from the near-body solution above must be equal to the drag from von Karman’s solution for the vortex street. To do this requires knowledge of the velocities and circulation in the wake. This can be determined in the same manner as described in the HPF model, following Roshko [14], and described below.

Only a fraction of the vorticity shed from the bluff body ends up in the wake. The rate at which vorticity is shed must be equal to rate of transport of circulation downstream in the vortex street. If $\epsilon$ is the fraction of shed vorticity that ends up in the vortex street, $\Gamma$ the strength of individual vortices, n the shedding frequency and $V_s$ is the velocity at separation, this can be written as

$$\begin{aligned} n\Gamma = \epsilon \frac{V_s^2}{2}. \end{aligned}$$

(15)

Rewriting n by using the freestream velocity, $V_\mathcal {Z}$, velocity of vortices relative to the freestream v, and the longitudinal spacing between individual vortices l, Eq. (15) becomes

$$\begin{aligned} \frac{V_\mathcal {Z} - v}{l} \Gamma = \epsilon \frac{V_s^2}{2}, \end{aligned}$$

(16)

$\Gamma$ can be removed from the equation by using one of von Karman’s stability parameters for a vortex street, $\frac{\Gamma }{vl} = 2\sqrt{2}$. Then

$$\begin{aligned} v = \frac{V_\mathcal {Z}}{2}\left( 1 \pm \sqrt{1 - \frac{\epsilon k^2}{2}}\right) . \end{aligned}$$

(17)

Here, $k_{bp}$ is the base pressure parameter, written as $k_{bp} = \frac{V_s}{V_\mathcal {Z}}$. Equation (17) relates the velocities in the wake with the freestream velocity.

The drag due to the vortex street, as shown by von Karman[10], is given by

$$\begin{aligned} C_d = \frac{h}{d}\left[ 5.65\left( \frac{v}{v_\infty }\right) - 2.25\left( \frac{v}{v_\infty }\right) ^2\right] . \end{aligned}$$

(18)

Here, h is the lateral dimension of the vortex street, given by the spacing between the shear layers from the conformal map in the $\mathcal {Z}$-plane, d is the reference length. Substituting in Eq. (18) for the wake velocity using Eq. (17), and using $v_\infty = V_\mathcal {Z}$, the equation for drag due to the vortex street becomes

$$\begin{aligned} C_d = \frac{h}{d}\left[ 1.70\left( 1 \pm \sqrt{1 - \frac{\epsilon k^2}{\sqrt{2}}}\right) + 0.563\frac{\epsilon k_{bp}^2}{\sqrt{2}}\right] . \end{aligned}$$

(19)

To join the near-body solution to the vortex street, the value of pressure drag calculated must be equal to the value of $C_d$ from Eq. (19). Equating drag allows calculating $\epsilon$ and then $\Gamma$.

Once $\Gamma$ is known, oscillating force due to vortex shedding can be calculated using a momentum approach, similar to Chen [15].

$$\begin{aligned} L' = \rho v\Gamma . \end{aligned}$$

(20)

Here, v is again the velocity of vortices in the vortex street relative to the freestream (all in the $\mathcal {Z}$-plane). Thus, a complete solution for flow around a bluff body is obtained.

2.4 Implementation

With the combination of Karman–Trefftz transformations and Fornberg’s method, a conformal map between an arbitrary geometry in the physical $\mathcal {Z}$-plane and a unit circle in the z-plane is calculated. To calculate flow around the geometry and subsequently forces due to vortex shedding and shedding frequency, the conformal map must be systematically used with the HPF model. The implementation of this procedure is outlined here:

• 1. The geometry in the physical ($\mathcal {Z}$) plane is first converted to a smooth closed shape in the $\zeta$-plane using multiple Karman–Trefftz transformations, written as a composition:

  $$\begin{aligned} \zeta = k_m \circ k_{m-1} \circ ... k_1 (\mathcal {Z}) = k(\mathcal {Z}). \end{aligned}$$

  KT maps can be explicitly inverted.

• 2. The Fornberg map between a unit disk centered in the z-plane and the smooth closed shape in the $\zeta$-plane is constructed:

  $$\begin{aligned} \zeta = h(z). \end{aligned}$$

• 3. In the z-plane, the streamlines (and particularly, the stagnation streamline) can be calculated using the Hybrid Potential Flow model.

• 4. Streamlines can now be mapped back from the circle z-plane to the physical $\mathcal {Z}$-plane. The stagnation streamline can similarly be mapped to the physical $\mathcal {Z}$-plane. In the HPF model, the stagnation streamline represents the surface of the cylinder prior to separation and the free shear layers in the wake after separation. Thus, by mapping back to the $\mathcal {Z}$-plane, the shape of the free shear layers there is obtained.

  $$\begin{aligned} \mathcal {Z} = k^{-1} \circ h(z). \end{aligned}$$

• 5. With flow in the physical $\mathcal {Z}$-plane now known, velocities and pressures can be computed over the geometry in question. The near-body model for the geometry is complete.

• 6. The vortex wake in the physical plane is represented by von Karman’s solution, and is joined to the near-body solution by equating the drag. Forces due to vortex shedding by the geometry can then be calculated.

This approach is outlined in Fig. 3.

Fig. 3

Overview of solution based on conformal mapping and potential flow theory

2.5 Applicable geometries

For any bluff body, a conformal map from the z-plane to the $\mathcal {Z}$-plane will always exist. However, realistic solutions using this technique are limited to a class of bluff bodies for which separation points on the cylinder map to points on the bluff bodies where flow is reasonably expected to separate (such as sharp corners). This class of bluff bodies is described below:

• 1. The bluff body must be closed and symmetric.

• 2. The rear stagnation point on the cylinder must map to a vertex. Thus, shapes such as a wedge or a semi-circular cylinder require the introduction of a very small artificial slope to allow a trailing edge vertex. An example of this is seen in Fig. 1.

• 3. Interior angles for corners must be between 0 and 180°.

• 4. Elongated bodies (such as a flat plate with some thickness at 0°angle of attack) will lead to separation points being mapped aft of actual separation points. Thus, fineness ratio of geometry can only be increased up to a point.

Thus, the bluff bodies for which this approach is tested are limited to symmetric bluff bodies of relatively small fineness ratios.

3 Results

3.1 Bluff body geometries

A combination of the HPF model and conformal mapping approach discussed in the previous section allows prediction of the vortex shedding behavior and Strouhal number for a range of bluff body geometries. Results for two bluff body geometries are presented here. These geometries were selected due to the easy availability of prior experimental results for validation. These geometries, a semi-circular cylinder and a 45°wedge, are shown in Figs. 4 and 5.

Fig. 4

Bluff body geometry used in conformal mapping: Semi-circular cylinder

Fig. 5

Bluff body geometry used in conformal mapping: 45°wedge

3.2 Flow in the physical $\mathcal {Z}$-plane

To calculate flow around each shape, the implementation procedure in Sect. 2.4 is followed. Flow in the circle (z) plane is computed at two Reynolds numbers in the sub-critical range, $Re_z = 1.7 \times 10^4$ and $Re_z = 6 \times 10^4$, using the Hybrid Potential Flow (HPF) model. This translates to a slightly different Reynolds numbers in the physical ($\mathcal {Z}$) plane, $Re_{\mathcal {Z}}$, depending on the bluff body geometry.

Here, Reynolds number is defined as:

$$\begin{aligned} Re = \frac{\rho v_\infty D}{\mu }, \end{aligned}$$

(21)

where standard atmospheric values for air are used for density ($\rho = 0.0024$ slug/ft$^3$), and dynamic viscosity ($\mu = 3.73 \times 10^{-7}$ lb-s/ft$^2$), $v_\infty$ is the freestream velocity. D is the characteristic length, which is the width of the bodies as shown in Figs. 4 and 5.

Streamlines around the semi-circular cylinder, calculated using the conformal map f, are shown in Fig. 6. Streamlines are first computed in the z-plane, then calculated in the $\zeta$-plane, before finally being computed in the $\mathcal {Z}$-plane. For this shape, $Re_\mathcal {Z} = 2.15 \times 10^4$.

Fig. 6

Calculation of flow around semi-circular cylinder. Here, $Re_z = 1.7 \times 10^4$ and $Re_\mathcal {Z} = 2.15 \times 10^4$. Shown are streamlines in all three planes

Table 1 Flow characteristics in the $\mathcal {Z}$-plane for $Re_z = 1.7\times 10^4$. In this table and Table 2, $Re_\mathcal {Z}$ is Reynolds number, $C_d$ the drag coefficient, $C_{l_{rms}}$ the rms lift coefficient, and St the Strouhal number, all in the $\mathcal {Z}$-plane. $Re_z$ is Reynolds number in the z-plane

Fig. 7

Calculation of flow around 45°wedge. Here, $Re_z = 1.7 \times 10^4$ and $Re_\mathcal {Z} = 2.45 \times 10^4$. Shown are streamlines in all three planes

Fig. 8

Calculation of flow around semi-circular cylinder. Here, $Re_z = 6 \times 10^4$ and $Re_\mathcal {Z} = 7.5 \times 10^4$. Shown are streamlines in all three planes

Fig. 9

Calculation of flow around 45°wedge. Here, $Re_z = 6 \times 10^4$ and $Re_\mathcal {Z} = 8.6 \times 10^4$. Shown are streamlines in all three planes

Table 2 Flow characteristics in the $\mathcal {Z}$-plane for $Re_z = 6\times 10^4$

The process is repeated for the 45°wedge, shown in Fig. 7. With wake width and velocities in the $\mathcal {Z}$-plane known, forces due to vortex shedding and Strouhal number can be computed. Flow characteristics for both shapes in the $\mathcal {Z}$-plane for $Re_z = 1.7 \times 10^4$ are shown in Table 1. Here, $Re_\mathcal {Z} = 2.45 \times 10^4$.

The process is also carried out for a higher Reynolds number in the circle plane, $Re_z = 6 \times 10^4$. In Figs. 8 and 9, the conformal mapping process for both shapes is shown again. The calculated Reynolds number in the $\mathcal {Z}$-plane, velocities in the wake, forces due to vortex shedding, and Strouhal number are also calculated and are shown in Table 2.

Calculating Strouhal numbers for both shapes in the sub-critical Reynolds number range allows comparison with published values. In Tables 3 and 4, predicted values for both shapes using this conformal mapping approach are compared with published values from Blevins [3].

Table 3 Comparison of predicted and published St numbers for semi-circular cylinder

Table 4 Comparison of predicted and published St numbers for 45°wedge

4 Conclusions and recommendations

The conformal mapping approach discussed in this paper, when used in tandem with the Hybrid Potential Flow model, allows easy prediction of vortex shedding behavior for bluff bodies of various geometries. Unlike previous theoretical approaches, this method allows the calculation not just of wake velocities, dimensions, and mean drag, but also the oscillating transverse force due to vortex shedding. Flow patterns and Strouhal numbers calculated for the bluff bodies considered correspond well to what is seen physically and previously published values. The systematic use of the HPF and conformal mapping technique can be an effective design tool to determine optimum geometries for use in small-scale energy devices that use VIV as its basis.

Further improvements can be made to the work presented here to make it more widely applicable. As previously mentioned, there are a class of bluff bodies to which this technique is applicable. Bluff bodies must be closed and symmetrical, and the separation points from the cylinder must map to points on the body where flow is reasonably expected to separate (such as sharp corners). To expand the class of bluff bodies for which this conformal mapping technique is applicable, including a uniqueness condition (such as the Kutta condition in Joukowski transforms) will be useful. In this manner, bluff bodies of larger fineness ratios can be used.

The HPF model and conformal mapping method outlined here predict vortex shedding frequencies for a rigid bluff body. If the body is elastic and undergoes large-amplitude VIV, then this motion can affect vortex patterns in the wake, which in turn can again affect body motion. Including the effect of body motion on the wake will result in a more complete model. Finally, comparing predicted loads due to vortex shedding for various bluff bodies with experimental measurements in a wind tunnel will further validate this method.