2.1. Fluid mechanical model

The three-dimensional fluid flow $\boldsymbol {u}$ is driven by the gradient of a pressure field $p$ and resisted by viscous stresses $\boldsymbol {\tau }$ ($=\mu ( \boldsymbol {\nabla } \boldsymbol {u} + (\boldsymbol {\nabla } \boldsymbol {u})^\textrm {T}$)). The dynamics of an incompressible Newtonian fluid are described by the differential forms of the conservation of momentum and mass:

(2.1a,b)\begin{equation} \rho \left(\frac{ \partial \boldsymbol{u}}{\partial t} + (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla}) \boldsymbol{u} \right) ={-} \boldsymbol{\nabla} p + \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{\tau}, \quad \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u} = 0, \end{equation}

where $\rho$ is the fluid density. The secondary flow depends on the presence of internal vorticity that comes from the form of the inlet flow and wall conditions. The velocity field is set on the pipe inlet through uniform or fully developed pipe flow analysis (see Appendix B) based on the actual pulsatile volume flux (see figure 2a),

(2.2)\begin{equation} Q(t) = \int u \,\mathrm{d}A = A U_b | \sin \varOmega_p t |, \end{equation}

and by imposing a kinematic constraint on the velocity at the pipe wall surface, where $U_b$ is the maximum bulk velocity, $\varOmega _p$ is the angular frequency and $A$ is the pipe cross-sectional area. On the pipe outlet surface, the pressure is set at $p=0$.

Figure 2.

(a) The variation of an inlet volume flux ($Q/U_b A$) with dimensionless time ($\tilde {t} = \varOmega _p t/ {\rm \pi}$) for one cycle. The four phases of one pumping cycle are impulsively starting ($Re \ll 1, \dot {Q}> 0$), accelerating ($Re \gg 1, \dot {Q}>0$), decelerating ($Re \gg 1, \dot {Q}<0$) and impulsively stopping ($Re \ll 1, \dot {Q}<0$). (b) A schematic of the computational domain along with the geometrical length scales. (c) The radial ($r$), azimuthal ($\theta$) and streamwise ($\phi$) coordinates. The total pipe bend surface is $S_b(=S_{bw}+S_{bi}+S_{bo})$ and includes the curved wall ($S_{bw}$), inlet ($S_{bi}$) and outlet ($S_{bi}$) surfaces. (d) The structured non-uniform mesh, shown for a slice normal to the pipe axis, is extruded along the pipe length. The coordinate transformation and non-dimensionlisation are discussed in Appendix A.

2.2. Key parameters

The pipe flow is characterised by the instantaneous Reynolds number ($Re$) and maximum bulk Reynolds number ($Re_b$), defined by

(2.3a,b)\begin{equation} Re = \frac{|u_m| D}{\nu}, \quad Re_b = \frac{t_{\nu }}{t_a} = \frac{U_b D}{\nu}, \end{equation}

respectively, where $t_{\nu }(=D^{2}/\nu )$ and $t_{a} (=D/U_b)$ are the viscous diffusion and advection time scales, respectively, and $\nu$ is the kinematic viscosity. The pulsatile flow (2.2) generates a velocity profile (see figure 13c), which has the contribution from both an oscillatory boundary layer, of approximate thickness

(2.4)\begin{equation} \delta_s = \sqrt \frac{2 \nu {\rm \pi}}{\varOmega_p}, \end{equation}

and steady core flow (characteristic thickness $\delta \sim D$). The relative size of the oscillatory boundary layer compared with the pipe diameter ($D/\delta _s$) gives an appropriate measure of the grid resolution required next to the wall. The Strouhal number expresses the ratio of advective time scale $t_a$ to oscillatory time scale $t_p (=2 {\rm \pi}/ \varOmega _p)$

(2.5)\begin{equation} St = \frac{t_a}{t_p}; \end{equation}

and is linked to $Re_b$ and $D/\delta _s$ through

(2.6)\begin{equation} \frac{D}{\delta_s} = \left(\frac{t_{\nu}}{t_p} \right)^{1/2} = St^{1/2} Re_b^{1/2} . \end{equation}

The steady to oscillatory boundary layer thickness scales approximate as $\delta /\delta _s \sim St^{1/2} Re_b^{1/2}$, so that as $St$ increases, the oscillatory boundary becomes thinner. The regime plot in figure 1 shows the iso-contours of $St$ for most numerical studies fall in the range $10^{-2} < St<10^{-1}$. The analysis in this study is focused on high-$St$ inertially dominated flows with most attention paid to $Re_b=2000$ and $St=1$.

The pipe geometry (figure 2b,c) consists of a $90^{\circ }$ bend connected with straight pipe sections. The curvature ratio $R_c/D$ is varied over the range of $1$ to $10$. The pipe bend is connected to the upstream and downstream pipes, whose lengths are $L_u/D (=3)$ and $L_d/D (=3)$, respectively. The total pipe length is $L_t = L_u + ({{\rm \pi} }/{2})R_c + L_d$.

2.3. Numerical implementation

Equations (2.1a,b) were solved on a well-resolved grid configuration using a well-validated finite-volume method within ANSYS FLUENT v19.0 (see supplementary material available at https://doi.org/10.1017/jfm.2021.553 for verification and validation). A time step $\Delta t$ was chosen from the constraint ($U_b \Delta t / h = 0.2$, where $h$ is the minimum grid size) to satisfy the Courant–Friedrichs–Lewy criterion for global stability.

3.1. Boundary layer analysis in an upstream straight pipe section

The first stepping stone is to analyse the unsteady boundary layer structure in the absence of a bend. We start with the assumption of an incompressible one-dimensional Navier–Stokes equation in the cylindrical coordinate system ($r,\theta , x$) along the axial direction $x$ (see figure 2b,c) in an infinitely long upstream pipe (uniform flow or axially invariant or fully developed pipe flow), with zero swirl ($u_\theta =0$, $u_r=0$, $u=u(r,t)$), cross-sectional spatially invariant pressure $p$ (${\partial p}/{\partial r} = 0$. Hence the axial pressure gradient is $\mathrm {d}p(x,t)/\mathrm {d}\kern0.7pt x$), and the boundary conditions (Das & Arakeri Reference Das and Arakeri2000) are

(3.1)\begin{equation} \left.\begin{gathered} \frac{\partial u}{\partial t} ={-} \frac{1}{\rho} \frac{\partial p}{\partial x} + \nu \left(\frac{\partial^{2} u}{\partial r^{2}} + \frac{1}{r} \frac{\partial u}{\partial r} \right), \\ u(D/2, t) = 0 \quad \mathrm{and} \quad \frac{\partial u(0,t)}{\partial r} = 0. \end{gathered}\right\}\end{equation}

When the pipe inlet volume flow rate is known,

(3.2)\begin{equation} \int_0^{D/2} 2 {\rm \pi}r u(r,t) \,\mathrm{d}r = \frac{\rm \pi}{4} D^{2} u_m(t) = Q(t), \end{equation}

where $r= r |_{\theta =0}=y-R_c$ for upstream straight pipe and $u_m(t)$ is the cross-sectional averaged velocity (or bulk velocity or flow rate). The streamwise boundary layer flow $u_\phi$ can be written as the sum of the steady and oscillatory components as

(3.3)\begin{equation} u_{\phi} = u_{\phi,s} + u_{\phi,o}. \end{equation}

Similarly, the azimuthal vorticity is composed of steady and oscillatory components:

(3.4)\begin{equation} \omega_{\theta} \left (={-} \frac{\partial u_\phi}{\partial r}\right) = \omega_{\theta,s} + \omega_{\theta,u} ={-} \frac{\partial u_{\phi,s}}{\partial r} - \frac{\partial u_{\phi,o}}{\partial r}. \end{equation}

For a pulsatile flow (2.2), the analytical boundary layer solution based on a given inlet volumetric flux is derived in Appendix B ((B9a,b), where $u_{\phi ,s} = u_1(r,t), \ u_{\phi ,o} = u_2(r,t)$). As $t \to \infty$, the non-dimensional form of $u_\phi$ (3.3) and $\omega _\theta$ (3.4) tend to

(3.5)\begin{align} {\tilde{u}_\phi}&= 2 \frac{\bar{u}_m }{U_b}(1-4 \tilde{r}_c^{2}) - \frac{1}{2} \frac{U_m}{U_b} \sum_{n=1}^{\infty} a_n \left( \frac{\textrm{J}_0(\gamma_a) - \textrm{J}_0(2 \tilde{r}_c\gamma_a)}{\textrm{J}_2(\gamma_a)} \exp({\textrm{i}n 2 {\rm \pi}St \tilde{t}})\right.\nonumber\\ &\quad\left. + \frac{\textrm{J}_0(\gamma_b) - \textrm{J}_0(2 \tilde{r}_c\gamma_b)}{\textrm{J}_2(\gamma_b)} \exp({-\textrm{i}n2 {\rm \pi}St \tilde{t}}) \right), \end{align}

(3.6)\begin{align} &\tilde{\omega}_{\theta} \left(={-} \frac{D}{U_b} \frac{\partial u_\phi}{\partial r}\right) \nonumber\\ &\quad =\! 16 \frac{\bar{u}_m}{U_b} \tilde{r}_c -\! \frac{U_m}{U_b} \sum_{n=1}^{\infty} a_n \left ( \frac{ \gamma_a \textrm{J}_1(2 \tilde{r}_c\gamma_a) }{\textrm{J}_2(\gamma_a)} \exp({\textrm{i}n 2 {\rm \pi}St \tilde{t}}) +\! \frac{ \gamma_b \textrm{J}_1(2 \tilde{r}_c\gamma_b) }{\textrm{J}_2(\gamma_b)} \exp({-\textrm{i}n 2 {\rm \pi}St \tilde{t}})\right)\!,\end{align}

where

(3.7a–c)\begin{equation} \tilde{r}_c = r/D, \quad\gamma_a = (n {\rm \pi}St Re_b)^{1/2} \frac{(1-i)}{\sqrt{2}}, \quad \gamma_b = (n {\rm \pi}St Re_b)^{1/2} \frac{(1+i)}{\sqrt{2}}. \end{equation}

3.2. Secondary flow in the pipe bend

Secondary flows can be analysed from a pressure–velocity $p - \boldsymbol {u}$ (Singh Reference Singh1974; Yao & Berger Reference Yao and Berger1975; Smith Reference Smith1976) or a vorticity $\boldsymbol {\omega }$ (Hawthorne Reference Hawthorne1951; Olson & Snyder Reference Olson and Snyder1985) perspective. Several new studies have categorised the secondary flow pattern under asymmetric pulsatile pipe flows (Komai & Tanishita Reference Komai and Tanishita1997; Boiron et al. Reference Boiron, Deplano and Pelissier2007; Krishna et al. Reference Krishna, Gundiah and Arakeri2017) using both approaches. For a starting–waiting flow (single positive pulse followed by a waiting period), both Boiron et al. (Reference Boiron, Deplano and Pelissier2007) and Krishna et al. (Reference Krishna, Gundiah and Arakeri2017) have interpreted the generation of a secondary flow from a $p - \boldsymbol {u}$ perspective; they describe its generation, which arises from the imbalance in the centrifugal force and the radial pressure gradient, causes the flow to be driven from the inner to the outer bend and the azimuthal return flow from the outer to inner bend. The $p - \boldsymbol {u}$ approach has an appealing physical interpretation but is complicated to solve analytically.

On the other hand, a $\boldsymbol {\omega }$ approach is generally more complicated to interpret but easier to solve analytically. Most recent studies have used vorticity or integral measures, such as circulation, as diagnostic tools to distinguish different types of secondary motions (Najjari et al. Reference Najjari, Cox and Plesniak2019). Kartik & Michael (Reference Kartik and Michael2018) used circulation to understand the morphological changes in large-scale coherent structures. Najjari et al. (Reference Najjari, Cox and Plesniak2019) looked at various terms in the vorticity equation to analyse Lyne-type vortex production and the influence of slip and no-slip wall conditions.

The generation of a secondary flow is associated with the streamwise component of vorticity ($\omega _\phi$) whose evolution with time is described by

(3.8)\begin{equation} \frac{\partial {\omega_\phi}}{\partial t} + (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla}) {\omega_\phi} = ({\boldsymbol{\omega} \boldsymbol{\cdot} \boldsymbol{\nabla}}) {u_\phi} + \nu {{\nabla}^{2} \omega_\phi}. \end{equation}

The first term on the right-hand side of (3.8) describes an inviscid stretching process, discussed by Hawthorne (Reference Hawthorne1951), Lighthill (Reference Lighthill1956) and Hunt (Reference Hunt1973), caused by the spatial gradients of $u_\phi$. This provides the source for the production of $\omega _\phi$:

(3.9)\begin{equation} S = (\boldsymbol{\omega} \boldsymbol{\cdot} \boldsymbol{\nabla}) u_{\phi} = \frac{\omega_\theta}{r} \frac{\partial u_\phi}{\partial \theta} + \omega_r \frac{\partial u_\phi}{\partial r} + \frac{\omega_\phi}{R_c} \frac{\partial u_\phi}{\partial \phi}, \end{equation}

through the tilting of the azimuthal and radial vorticity ($\omega _\theta$, $\omega _r$) by cross-stream variations of $u_{\phi }$ (first two terms of 3.9) and the compression of $\omega _\phi$ by a streamwise variation of $u_\phi$ (last term of (3.9)).

The analyses by Dean (Reference Dean1928) and Lyne (Reference Lyne1971) are based on small-curvature pipes ($R_c/D \gg 1$) where the cross-stream strain is approximated by

(3.10)\begin{equation} \boldsymbol{\nabla} u_\phi = \frac{\gamma}{R_c} \frac{\partial u_\phi}{\partial y^{\prime}} \hat{\boldsymbol{c}}, \end{equation}

($\gamma =2$ and $\boldsymbol {\hat {c}}$ is the cross-wise unit vector defined in figure 2c) and this leads to a simplification of the source term as

(3.11)\begin{equation} S = (\boldsymbol{\omega}\boldsymbol{\cdot} \boldsymbol{\nabla}) u_{\phi} = \frac{\gamma}{R_c} (\omega_\theta {u_\phi} \sin\theta + \omega_r {u_\phi} \cos\theta). \end{equation}

Dean (Reference Dean1928, (11)) and Lyne (Reference Lyne1971, (2.14)) studied the secondary flow generated in an infinitely long pipe, where both sources in (3.11) are important. For an infinitely long pipe, the self-induced motion of the secondary flow is taken into account in the advection term, but streamwise advection is neglected (because $\partial / \partial \phi =0$). However, near the bend inlet (where $|\omega _r| \ll |\omega _\theta |$), the dominant source term arises from stretching of $\omega _\theta$,

(3.12)\begin{equation} S \simeq \frac{\gamma}{R_c} \omega_\theta u_\phi {\sin\theta}, \end{equation}

and $\omega _\phi$ is primarily advected by the streamwise flow. This leads to a simplification and linearisation of (3.8):

(3.13)\begin{equation} \frac{\partial {\omega_\phi}}{\partial t} + u_\phi \frac{\partial \omega_\phi }{\partial s} = \frac{\gamma \omega_\theta u_\phi}{R_c} \sin \theta + \nu {{\nabla}^{2} \omega_\phi}. \end{equation}

The simplicity of this linear model allows the contributions from the inlet azimuthal vorticity, steady core and oscillatory boundary layers to $\omega _\phi$ to be understood and quantified. As $\phi$ increases, the self-induced motion created by the secondary flow becomes more important, and its influence needs to be taken into account.

The steady inviscid result of Hawthorne (Reference Hawthorne1951, (12)) and Olson & Snyder (Reference Olson and Snyder1985, (3)) can be recovered from (3.13) by noting that the streamwise vorticity after a distance $s$, along the pipe, is

(3.14)\begin{equation} \omega_\phi \simeq \int_0^{s} \frac{S}{u_\phi} \mathrm{ d}s \simeq \gamma \phi \omega_{\theta i} \sin\theta, \end{equation}

where $\omega _{\theta i}$ is the inlet azimuthal vorticity field. For a pulsatile flow, the source term averaged over a period of $2{\rm \pi} /\varOmega$ is

(3.15)\begin{equation} \bar{S} = \frac{\gamma}{R_c} (P_{ss} + P_{oo}) \sin \theta, \end{equation}

(where the overbar represents a time average) which has separate contributions from the steady core and oscillatory components of the flow:

(3.16a,b)\begin{equation} P_{ss}={\omega_{\theta ,s} u_{\phi ,s}}, \quad P_{oo}=\overline{\omega_{\theta ,o} u_{\phi ,o}}. \end{equation}

The numerical solution to (3.13) is discussed in Appendix C.

The linear model provides a framework to understand how the secondary flow grows in the near-bend inlet region and the distinct contributions from steady and oscillatory flow components to $\omega _\phi$. The framework extends Hawthorne's (Reference Hawthorne1951) inviscid analysis to include viscous diffusion and distinguishes the distinct contributions from inviscid stretching and the no-slip condition to $\omega _\phi$. Because the model neglects nonlinear processes, such as self-induced motion and stretching of $\omega _r$, its validity is limited to the near-bend inlet region and small-curvature pipes.

3.3. Forces

The force on the pipe bend (A4) can be calculated directly from the integral of the pressure and viscous stresses over $S_{bw}$. It is more insightful to re-express the force in terms of an integral over the pipe bend inlet and outlet surfaces ($S_{bi}$ and $S_{bo}$) (see figure 2c), and the pipe bend volume $V$ using the momentum integral approach (Tay & Thorpe Reference Tay and Thorpe2014; Potter, Wiggert & Ramadan Reference Potter, Wiggert and Ramadan2016; White Reference White2016; Yunus Reference Yunus2017), to give

(3.17)\begin{equation} \boldsymbol{F}_p ={-} \int_{S_{bi} + S_{bo}} p \hat{\boldsymbol{n}}\, \mathrm{d} S - \rho \left ( \int_V \frac{\partial \boldsymbol{u}}{\partial t} \mathrm{d} V + \int_V (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla}) \boldsymbol{u} \,\mathrm{d} V \right) + \int _{S_{bi} + S_{bo}} \boldsymbol{\tau} \boldsymbol{\cdot} \hat{\boldsymbol{n}}\,\mathrm{d} S. \end{equation}

When the outlet surface ($S_{bo}$) is at an angle $\phi$ with respect to the inlet surface ($S_{bi}$) the force integral can be written as

(3.18)\begin{align} \boldsymbol{F}_p \left (= (F_x, F_y) \right) &={-} \left ( p_{bi} A + \rho \int_{S_{bi}} u_\phi^{2} \,\mathrm{d}A + \rho \int _{S_{bo}} u_\phi v' \,\mathrm{d}A \right) \hat{\boldsymbol{n}}_i \nonumber\\ &\quad - \left ( p_{bo} A + \rho \int_{S_{bo}} {u_\phi}^{2} \,\mathrm{d}A + \rho \int _{S_{bi}} u_\phi v' \,\mathrm{d}A \right)\hat{\boldsymbol{n}}_o \nonumber\\ &\quad - \rho \int_V \frac{\partial \boldsymbol{u}}{\partial t} \mathrm{d}V - \int_{S_{bi} + S_{bo}} \boldsymbol{\tau} \boldsymbol{\cdot} \hat{\boldsymbol{n}} \,\mathrm{d}S, \end{align}

where $p_{bi}$ and $p_{bo}$ are the bend inlet and outlet pressures, and $\hat {\boldsymbol {n}}_i (=(-1,0) )$, $\hat {\boldsymbol {n}}_o(=(\cos \phi ,-\sin \phi ) )$ are unit vectors normal to the pipe bend inlet and outlet. The $F_x$ is quite different from the $F_y$ expression. We anticipate, and indeed find, that $F_x$ and $F_y$ are different. Tay & Thorpe (Reference Tay and Thorpe2014) and Xiao et al. (Reference Xiao, Jing, Han, Yang and Wang2019) applied a momentum approach to evaluate the unsteady forces on a pipe bend arising from slugs passing, where the fluid volume ($V$) changed with time owing to the slug movement, but the volume flux ($Q$) was constant. In our example, the fluid volume ($V$) is constant, but volume flux ($Q$) changed with time.

The velocity field in a curved pipe can be written as the summation of three components – streamwise ($\hat {\boldsymbol {s}}$), cross-wise ($\boldsymbol {\hat {c}}$) and spanwise ($\hat {\boldsymbol {a}}$) (defined in figure 2c), as

(3.19a,b)\begin{equation} \boldsymbol{u} = u_{\phi} \hat{\boldsymbol{s}} + v^{\prime} \boldsymbol{\hat{c}} + w^{\prime} \hat{\boldsymbol{a}}, \quad \frac{\mathrm{d} \boldsymbol{u}}{\mathrm{d}t} = \frac{\mathrm{d} {u}_{\phi}}{\mathrm{d}t} \hat{\boldsymbol{s}} + \frac{\mathrm{d} {v^{\prime}}}{\mathrm{d}t} \boldsymbol{\hat{c}} + \frac{\mathrm{d} {w^{\prime}}}{\mathrm{d}t} \hat{\boldsymbol{a}}, \end{equation}

where the tangential unit vector is $\hat {\boldsymbol {s}} (=(\cos \phi ,-\sin \phi ) )$. The velocity components scale as $u_{\phi } \sim u_m$ and $v^{\prime }, w^ \prime \sim \lambda u_m$. The typical values of $\lambda$ are $\lambda < 1$ and $\lambda \ll 1$ for $D/\delta _s \geqslant 0$ and $D/\delta _s \gg 0$, respectively. This means that the force arising from the secondary flow is small compared with the contribution from the streamwise components. Therefore,

(3.20) \begin{align} \boldsymbol{F}_p &= \underbrace{- \left(\bar{p}_{bi} + \rho \overline{u_{\phi i}^{2}} \right) A \hat{\boldsymbol{n}}_i - \left (\bar{p}_{bo} + \rho \overline{u_{\phi o}^{2}} \right) A \hat{\boldsymbol{n}}_o - \rho \frac{\mathrm{d} u_m}{\mathrm{d}t} \int_V \hat{\boldsymbol{s}} \,\mathrm{d}V}_{\textit{streamwise components}} \nonumber\\ &\quad - \underbrace{ \rho A \left ( \overline{{u_{\phi i} v_i^{\prime}}} \hat{\boldsymbol{n}}_i + \overline{{u_{\phi o} v_o^{\prime}}} \hat{\boldsymbol{n}}_o \right) - \rho \int_V \frac{\mathrm{d} ( v^{\prime} \boldsymbol{\hat{c}} + w^{\prime} \hat{\boldsymbol{a}})}{\mathrm{d}t} \mathrm{d}V} _{\textit{cross-wise components}} - \int_{S_{bi} + S_{bo}} \boldsymbol{\tau} \boldsymbol{\cdot} \hat{\boldsymbol{n}} \,\mathrm{d}S, \end{align}

where the overbar represents the cross-sectional averaged quantities. The volume flux across the pipe is constant along the pipe length owing to incompressibility, and hence both $\int u_{\phi } \,\mathrm {d}A$ and $({\mathrm {d}}/{\mathrm {d}t})\int u_{\phi } \,\mathrm {d}A$ are independent of $\hat {\boldsymbol {s}}$. So, we can set

(3.21a,b)\begin{equation} \bar{u}_{\phi o} = \bar{u}_{\phi i} = u_m(t), \quad \frac{\mathrm{d} \bar{u}_{\phi o}}{\mathrm{d}t} = \frac{\mathrm{d} \bar{u}_{\phi i}}{\mathrm{d}t} = \frac{\mathrm{d} {u}_{m}}{\mathrm{d}t}. \end{equation}

Additionally,

(3.22a,b)\begin{equation} \rho \overline{u_{\phi i}^{2}} = \gamma_i \rho u_m^{2}, \quad \rho \overline{u_{\phi o}^{2}} = \gamma_o \rho u_m^{2}, \end{equation}

where $\gamma =\overline {u_{\phi }^{2}}/u_m^{2}(t)$ depends on the velocity profile across the pipe. The dominant terms in (3.20) are identified in the limit of high-$Re_b$. The secondary flow generates cross-wise momentum flux whose terms scale as $\rho \lambda u_m^{2} A$, and because $\lambda \ll 1$, their contribution to the total bend force is small. The contribution from the secondary flow acceleration is expected to scale as $\rho \lambda A R_c u_m \varOmega$, and again, is smaller than the contribution from the streamwise flow acceleration and can be negligible. The viscous contribution arises from the integration over the inlet and outlet planes and therefore scales as $\mu u_m D \sim \rho u_m^{2} A/Re_b$. As $Re_b$ is large, the viscous contribution term is negligible.

At high $Re_b$, the main contributions to the bend force are from the deflection of the pipe flow arising from the change in $\hat {\boldsymbol {n}}$ and the integral along the bend (from $\int \hat {\boldsymbol {s}} \,\textrm {d} V$), with weaker contributions from the secondary flow and viscous effects. The pipe bend force is, to leading order,

(3.23)\begin{equation} \boldsymbol{F}_p \simeq{-}(\bar{p}_{bi} + \gamma_i \rho u_m^{2}) A \hat{\boldsymbol{n}}_i -(\bar{p}_{bo} + \gamma_o \rho u_m^{2}) A \hat{\boldsymbol{n}}_o - \rho \frac{\mathrm{d} u_m}{\mathrm{d}t} \int_V \hat{\boldsymbol{s}} \,\mathrm{d}V. \end{equation}

For a pipe with a bend angle of $\phi _m$, the pipe bend forces in the $x$- and $y$-directions reduce to

(3.24)\begin{equation} \left.\begin{gathered} {F}_x = \left( \bar{p}_{bi} - \bar{p}_{bo} \cos\phi_m + \gamma_i \rho u_m^{2} (1 - \cos\phi_m) - \rho \frac{\mathrm{d} u_m}{\mathrm{d}t} R_c \sin\phi_m \right) A , \\ {F}_y = \left ( (\bar{p}_{bo} + \gamma_o \rho u_m^{2} ) \sin\phi_m - (\cos\phi_m -1) \rho \frac{\mathrm{d} u_m}{\mathrm{d}t} R_c \right) A. \end{gathered}\right\} \end{equation}

In the limit of high $Re_b$ and high $St$, the bend force can be estimated by noting that the pressure variation along the whole pipe is dominated by the flow acceleration, and the pressure variation across the pipe is small ($\bar {p}_{bi} \simeq p_{bi},\bar {p}_{bo} \simeq p_{bo}$). When the exit pipe pressure is $p_e$, this shows that the pressure at a point in the pipe is proportional to the distance to the tube outlet $s_o$, i.e.

(3.25)\begin{equation} p = p_e+ \int_{L}^{s} \boldsymbol{\nabla} p \boldsymbol{\cdot} \mathrm{d}\hat{\boldsymbol{s}} = p_e - \rho \frac{\mathrm{d} u_m}{\mathrm{d} t} (s-L). \end{equation}

The distance from the pipe bend exit to the pipe exit is $s_{o}=L_d$, so the distance from the pipe bend inlet to the pipe exit is $s_{i} = \phi _m R_c + L_d$. For a $90^{\circ }$ pipe bend ($\phi _m={\rm \pi} /2$), the difference between bend force in the $x$- and $y$-directions is

(3.26)\begin{align} F_x - F_y &= \left ( \bar{p}_{bi} - \bar{p}_{bo} + (\gamma_i - \gamma_o) \rho u_m^{2} - 2 \rho \frac{\mathrm{d}u_m}{\mathrm{d}t} R_c \right) A \nonumber\\ &= \left( - \rho R_c \left( \frac{\rm \pi}{2} + 2 \right) \frac{\mathrm{d}u_m}{\mathrm{d}t} + (\gamma_i - \gamma_o) \rho u_m^{2} \right ) A, \end{align}

and arises mainly from the pressure drop along the pipe and the acceleration force (because typically $\gamma _0 \simeq \gamma _i$). For a pulsatile flow (2.2) driven around a $90^{\circ }$ bend ($\phi _m={\rm \pi} /2$), the $x$-component of the force over one cycle of the flow is

(3.27)\begin{equation} F_x = \rho A U_b^{2} \left (- St \frac{2 {\rm \pi}(L_d+R_c({\rm \pi}/2 + 1))}{D} \cos \varOmega_p t + \gamma_i \sin^{2} \varOmega_p t \right) + p_e A , \quad \mathrm{for} \ 0 \leqslant \varOmega_p t \leqslant {\rm \pi}. \end{equation}

From (3.27), the influence of the flow acceleration arises from the first term, which dominates over the force owing to the momentum flux when

(3.28)\begin{equation} St > \underbrace{\frac{D}{2{\rm \pi}(L_d + {\rm \pi}R_c/2 + R_c)}}_{St_{cr}}. \end{equation}

As the length of pipe after the bend increases, the pressure in the bend region increases in magnitude and makes a greater contribution to the total bend force. The impact of the pipe length $L_d$ on the bend force simply arises from the pressure increasing linearly along the pipe and being proportional to the flow acceleration. The simple analysis shows that the force depends on the length of the outlet pipe for high-$St$ flows.

4.1. Velocity field

The three-dimensional flow streamlines are shown in figure 3(a–d), and a colour map has been added to them, which indicates the flow speed. The accelerating ($\tilde {t}=0.25$) and decelerating ($\tilde {t}=0.75$) phases show a similar streamline pattern as the steady flow simulation with the core flow skewed towards the outer bend region. A helical streamline pattern is seen after the bend, which is associated with the streamwise vorticity. This pattern becomes more pronounced for the pulsatile case when $\tilde {t}=0$ because the mean flow is smaller along the pipe.

Figure 3.

Comparison of three-dimensional streamlines in (a) a steady inlet flow ($Re_b = 2000$ and $St=0$) and (b–d) a pulsatile inlet flow ($Re_b =2000$ and $St = 1$) in a pipe bend ($R_c/D=1.5$) corresponding to (b) $\tilde {t}= 0$, (c) $0.25$ and (d) 0.75. (e) Shear stress patterns on the outer wall and inner wall for steady and pulsatile inlet flow conditions.

There is extensive literature on three-dimensional flow separation – centred around the topological shear stress pattern (Hunt et al. Reference Hunt, Abell, Peterka and Woo1978; Tobak & Peake Reference Tobak and Peake1982), distinguishing between flow reversal and separation (Das & Arakeri Reference Das and Arakeri1998), and spatial/temporal pressure gradients (Das, Srinivasan & Arakeri Reference Das, Srinivasan and Arakeri2016; Krishna et al. Reference Krishna, Gundiah and Arakeri2017). Here, the point of flow separation is discussed through viscous stress lines on the pipe surface, which obeys certain topological rules (see Hunt et al. Reference Hunt, Abell, Peterka and Woo1978). Figure 3(e) shows the shear stress lines (with $|\tilde {\boldsymbol {\tau }}|$ mapping) on the pipe surface for steady and pulsatile inlet flow conditions and the locations of singular points (saddle and nodes) in the stress field are indicated. The surface of separation is evident for the steady inlet flow simulation by a saddle point in the near-bend outer wall region (upstream); it can also be identified from a detailed examination of the local streamlines. Flow separation is observed during the decelerating phase ($\tilde {t}=0.75$) and is identified by a saddle point after the bend (in the outer wall region). While no separation is observed at $\tilde {t}=0$, flow reversal occurs adjacent to the pipe wall owing to the velocity profile inflection at $\tilde {t}=0$ (see figure 13c). During the accelerating phase ($\tilde {t}=0.25$), the flow is accelerating everywhere, which suppresses the potential for separation.

For a steady inlet flow, the consequence of high $Re_b$ and large curvature is that the flow impacts the bend outer wall generating a weak return flow (in the form of an upstream perturbation), which leads to a singular point and, as a consequence, flow separation upstream. These are evident as a node in the surface shear stress pattern in figure 3(e). Smith (Reference Smith1976) identified an anomalous pressure gradient upstream of the pipe bend and on the outer wall in a high-$Re_b$ ($Re_b \gg 1$) small-curvature ($R_c/D \gg 1$) pipe flow. This perturbation appears to be significant enough at large curvature to generate separation, as observed in figure 3(e). This effect weakens at low $Re_b$ and small curvature, where the saddle point (in the upstream) shifts to the bend region and becomes closer to the node. Weak flow reversal and the upstream perturbation seem to appear in Krishna et al. (Reference Krishna, Gundiah and Arakeri2017, figure 10c).

Figure 4 shows a comparison between the velocity profiles for steady and pulsatile flows. Upstream of the pipe bend, the streamwise velocity profiles are symmetric. In the near-bend region, the velocity profiles for the pulsatile flow case have a similar shape over time but are laterally displaced depending on the instantaneous volume flux. With increasing $\phi$ ($\phi >30^{\circ }$), the flow is skewed towards the outer wall. For a steady inlet flow, the weak negative velocities near the bend inlet (outer wall region) arise from flow separation (see figure 4(a) that can be identified by the saddle point in figure 3e). It is worth noting that while the streamwise velocity is positive at the inlet, the effect of the viscous boundary layer leads to flow reversal and a strong gradient of velocity at the wall when $\tilde {t}= 0$ and $u_m = 0$. This is especially prominent in figure 4(b), with no evidence of flow separation seen in figure 3(e).

Figure 4.

Profiles showing the instantaneous streamwise velocity in a plane $z=0$ along $y^{\prime }$ at various bend ($\phi =0^{\circ }$ to $90^{\circ }$) and upstream locations. The results are for (a) a steady inlet flow ($Re_b = 2000$ and $St=0$) and (b,c) a pulsatile inlet flow ($Re_b =2000$ and $St = 1$) in a pipe bend ($R_c/D=1.5$) corresponding to (b) $\tilde {t}= 0$ and (c) $0.25$. Upstream data were captured at a distance of $1.5D$ from the bend inlet.

The time-averaged streamwise velocity $\bar {\tilde {u}}_\phi$ along $\tilde {y}^{\prime }$ (the line extending from the inner to the outer wall) and $\tilde {z}^{\prime }$ (the line extending from the bottom to top sidewall) at bend locations are shown in figure 5(a,b), respectively. The variation of $\bar {\tilde {u}}_\phi$ along $\tilde {z}^{\prime }$ shows that the streamwise velocity is approximately parabolic in the near-bend inlet region ($\phi <30^{\circ }$).

Figure 5.

Variation of time-averaged streamwise velocity $\overline {\tilde {u}}_{\phi }$ along (a) $\tilde {y}^{\prime }$ and (b) $\tilde {z}^{\prime }$ at various bend locations ($\phi =0^{\circ }$, $15^{\circ }$, $30^{\circ }$, $60^{\circ }$ and $90^{\circ }$). The results are for a pulsatile inlet flow ($Re_b =2000$ and $St = 1$) in a pipe bend ($R_c/D=1.5$).

4.2. Vorticity field

4.2.1. Azimuthal vorticity

In a straight pipe section, the azimuthal vorticity for a pulsatile inlet flow condition has two distinct components (3.4) – a steady component ($\tilde {\omega }_{\theta ,s}$) in the steady boundary layer (thickness, $\delta$) and an oscillatory component ($\tilde {\omega }_{\theta ,u}$) in the Stokes boundary layer (thickness, $\delta _s$). In figure 6(a), the linear model of $\tilde {\omega }_\theta$ (3.4) consists of a slow decay (negative) associated with the steady boundary vorticity (over distance $\delta$) and rapid oscillatory decay (over distance $\delta _s$). These analytical results capture quite well the numerical results at $1.5D$ upstream of the pipe inlet (see figure 6a). The effect of diffraction and distortion of $\tilde {\omega }_{\theta }$ can be seen in the bend region at $\phi =30^{\circ }$, where the profiles of $\tilde {\omega }_{\theta }$ change (see figure 6b).

Figure 6.

The dimensionless azimuthal vorticity ($\tilde {\omega }_\theta$) for a pulsatile inlet flow condition ($Re_b = 2000, St=1$ and $R_c/D= 1.5$) is plotted as a function of the distance from the sidewall along the line $\theta = 90^{\circ }$ for times $\tilde {t} = 0$, 0.25, 0.50 and $0.75$. In (a), the results are plotted in the straight upstream pipe – numerically (solid lines) at $1.5D$ upstream of the bend and analytically (circular markers) based on (3.4). In (b), the numerical results are shown for the pipe-bend region at $\phi = 30^{\circ }$.

4.2.2. Source of streamwise vorticity ($\bar {S}$)

The time-averaged source of $\omega _\phi$, $\bar {S}$, is proportional to the sum of $\tilde {P}_{ss}$ and $\tilde {P}_{oo}$, which are proportional to the production of the streamwise vorticity from the steady and oscillatory components of the flow, and these are discussed in relation to figure 7. Figure 7(a) shows a comparison of the average of $\tilde {P}={\tilde {\omega }_\theta \tilde {u}_\phi }$ over one period for a range of $St$ spanning steady and pulsatile flows. This shows that the source term (which is proportional to $\tilde {P}$) for steady and high-Strouhal-numbers are similar except within the thin Stokes layer.

Figure 7.

Spatial distribution of dimensionless time-averaged production terms plotted against the distance from the sidewall along the line $\theta = 90^{\circ }$. (a) A comparison of time-averaged $\bar {\tilde {P}}=\overline {\tilde {\omega }_\theta \tilde {u}_\phi }$ (3.16a,b) over one period for an inlet source is plotted for $St$ varying from steady inlet flow ($Re_b=2000$ and $St=0$) to high-$St$ limit ($Re_b=2000$ and $St=1$). (b) The numerical results (solid lines) upstream of the bend (captured at $1.5D$) are compared with the analytical results (circular markers). The numerical results are also shown in the bend region at (c) $\phi =15^{\circ }$ and (d) $\phi =30^{\circ }$. The results in (b–d) are for a pulsatile inlet flow ($Re_b =2000$ and $St = 1$) in a pipe bend ($R_c/D=1.5$).

Figure 7(b) shows a comparison between $\tilde {P}$ evaluated from the numerical flow simulations and the analytical predictions. The observations confirm the slow variation of the single signed $\tilde {P}_{ss}$ over $\delta$, while $\tilde {P}_{oo}$ is larger in magnitude but decays rapidly over distance $\delta _s$ from the wall. The total production term ($\tilde {P}=\tilde {P}_{ss}+\tilde {P}_{oo}$) has a small negative dip beyond the maximum. The total production term is numerically evaluated (using (3.16a,b)) at $1.5D$ upstream of the bend (quite consistent with the analytical model; see figure 7b) and at bend locations $\phi =15^{\circ }$ (see figure 7c) and $\phi =30^{\circ }$ (see figure 7d). The decomposition of $\tilde {P}$ into $\tilde {P}_{ss}$ and $\tilde {P}_{oo}$ exploited the slow change in $\tilde {P}_{ss}$ and $\tilde {P}_{oo} (=\tilde {P}- \tilde {P}_{ss})$. The essential character of the analytical model is confirmed.

4.2.3. Streamwise vorticity

Figure 8 shows the variation of the streamwise vorticity $\tilde {\omega }_\phi$ at various times ($\tilde {t}$) and bend positions ($\phi$). The $\tilde {\omega }_\phi$ profiles (see figure 8a) in the lower sidewall bend region are negative in the core region (consistent with (3.15)) because the source term is negative in the core and positive near the wall owing to production via the no-slip condition (see figure 7b–d). Owing to high $St$, the upstream pipe flow consists of a steady distribution of $\tilde {\omega }_\theta$ within the pipe core and an oscillatory boundary layer confined near the wall (see figure 6a). While $\tilde {u}_\phi$ is unsteady, its profile is unchanged in the core region and simply moves backwards and forwards (see figure 4b,c), quite different from the low-$St$ scenario. This results in the source term from stretching $S$ that has a smaller in-cycle variation. The first observation is that $\tilde {\omega }_\phi$ tends to be time-independent (steady) in the core region (see figure 8a). The second observation is that at high $St$, $\tilde {\omega }_\phi$ has a close correspondence (in the core region) with the case $St=0$ (see figure 8b). While Sumida et al. (Reference Sumida, Sudou and Wada1989) pointed out the similarity of the cross-flow streamlines, the range of $St$ was limited to identify the relationship between the low- and moderately-high-$St$ cases. The origin of the weak dependence of $\tilde {\omega }_\phi$ on time and the similarity between $St=0$ and $St=1$ become clear once the production terms are compared (see figure 7a).

Figure 8.

Variation of $\tilde {\omega }_\phi$ with distance from the sidewall along line $\theta = 90^{\circ }$. (a) Instantaneous ${\tilde {\omega }}_\phi$ profiles for pulsatile inlet flow case ($Re_b=2000$, $St=1$ and $R_c/D=1.5$) at a bend location $\phi = 15^{\circ }$ and various times ($\tilde {t} = 0$, 0.25, 0.5, and $0.75$). (b) Comparison of $\tilde {\omega }_\phi$ between steady ($Re_b=2000$ and $St=0$) and pulsatile ($Re_b=2000$ and $St=1$) inlet flow conditions. (c) Time-averaged $\bar {\tilde {\omega }}_\phi$ profiles at various bend locations ($\phi = 0^{\circ }$, $15^{\circ }$, $30^{\circ }$, $60^{\circ }$ and $90^{\circ }$) based on numerical simulation results. (d) The $\bar {\tilde {\omega }}_\phi$ generation (based on linear model (3.13) and (C1)) from stretching ($\tilde {\hat {\omega }}_I$) and no-slip conditions ($\tilde {\hat {\omega }}_H$) at $\phi =15^{\circ }$. The inset in (d) shows a comparison of $\bar {\tilde {\omega }}_\phi$ between steady ($Re_b=2000$ and $St=0$) and pulsatile ($Re_b=2000$ and $St=1$) inlet flow conditions.

Figure 8(c) shows a comparison of the $\bar {\tilde {\omega }}_\phi$ profiles at various bend locations. Beyond $\phi =30^{\circ }$, the self-induced flow caused by the streamwise vorticity leads to the rearrangement of the streamwise vorticity (see figure 9a). Figure 8(d) shows the predictions by the linear model (3.13) highlighting the distinct contributions for the vorticity generated by stretching ($\tilde {\hat {\omega }}_{I}$) and from the no-slip ($\tilde {\hat {\omega }}_{H}$) condition on the wall. The critical element is that shear in the streamwise flow enhances cross-stream diffusion, which amplifies the streamwise vorticity ($\omega _{\phi }$). This diffusive process is absent from the analysis by Hawthorne (Reference Hawthorne1951). A comparison is shown between the $\bar {\tilde {\omega }}_\phi$ prediction based on $St=0$ and $St=1$ (see inset in figure 8d). The comparison shows that the core flow distribution is identical, and the differences only occur near the wall. The linear model, based on small-curvature ($R_c/D \gg 1$), provides a qualitative view of the numerical simulations, which are based on large-curvature ($R_c/D=1.5$).

Figure 9.

(a) Evolution of streamwise vorticity ($\bar {\tilde {\omega }}_\phi$) along pipe bend with various cut-slices of flow, along with a snapshot at $\phi = 90^{\circ }$. (b) Instantaneous vortical core structures using an iso-surface of the second invariant (A3) ($\widetilde {II} = 5.8 \times 10^{-3}$) with the velocity magnitude $|\tilde {\boldsymbol {u}}|$ plotted over iso-surfaces. The result is for a pulsatile flow with $Re_b =2000$ and $St = 1$ corresponding to $\tilde {t}= 0$.

Figure 9(a) shows the cross-stream distribution of $\bar {\tilde {\omega }}_\phi$ in several planes along the bend length for a pulsatile inlet flow at $\tilde {t}=0$. (Note: vorticity distribution at all times $\tilde {t}=0$, 0.25, 0.5, 0.7 show similar characteristics as steady, and therefore only the results of $\tilde {t}=0$ are shown here.) An iso-surface of the second invariant is plotted (see figure 9b) to differentiate the contribution of secondary motion from the mean (steady) and an oscillatory boundary layer in the bend region. The transition from a pure sidewall secondary vortex pair (in the near-bend region at $\phi =0^{\circ }$ to $15^{\circ }$) to the deformed secondary vortex pair in the inner bend (at $\phi =90^{\circ }$) is observed in both a steady and pulsatile inlet flow. The Stokes region of a pulsatile inlet flow shows an additional near-wall coherent structure throughout the inner bend region, which is generated owing to the strong negative velocity gradient at the impulsively starting ($\tilde {t}=0$) and stopping phase ($\tilde {t}=1$).

5.1. Viscous shear stress

The viscous shear stress is proportional to the vorticity on the wall and provides an idea of the viscous component of the force. Figure 10 shows the variation of the shear stress ($\tilde {\tau }_{\phi r}$) at a point in an inner (see figure 10a) and outer (see figure 10b) bend region over $\tilde {t}=0$ and $1$. These are captured for the first cycle ($\varOmega _p t/ {\rm \pi}$) and 51st cycle ($\varOmega _p t/ {\rm \pi}- 50$) when the initial transient is lost. A comparison is made with the analysis of Komai & Tanishita (Reference Komai and Tanishita1997), which uses a large-curvature pipe model to analyse flow adjustment over the initial cycle, and reasonably captures our observations of a strong wall shear stress at the inner bend. The maximum shear stress at the outer bend is $35\,\%$ (during the first cycle) and $40\,\%$ (during the 51st cycle) of that at the inner bend. Krishna et al. (Reference Krishna, Gundiah and Arakeri2017) also showed a similar difference in the wall shear stress of $50\,\%$ (ratio: outer to inner bend) during the first cycle for a square cross-sectional curved pipe.

Figure 10.

Variation of the dimensionless wall shear stress ($\tilde {\tau }_{\phi r}$) over one period, which contrasts the values on the (a) inner and (b) outer walls at a bend location $\phi = 45^{\circ }$. The pulsatile parameters are $Re_b = 2000$ and $St=1$, and $R_c/D = 1.5$. The shear stress in a strongly curved pipe calculated by Komai & Tanishita (Reference Komai and Tanishita1997) is plotted with circular markers.

5.2. Pressure

The pipe flow is alternately accelerated and decelerated along the pipe, and the pressure is expected to vary linearly with distance at high-$St$ (3.25). Figure 11(a) shows the pressure ($\tilde {p}_s$) variation over a pipe length along the inner surface, centreline, and outer surface, plotted against the centreline distance ($\tilde {s}$) from the pipe inlet, which confirms the proportionality of $p$ to $\partial u_m / \partial t$ and $s$ (3.25). The shaded region represents the extent of the bend region. The pressure gradient changes sign during one cycle, but the non-dimensional form in figure 11(a) is positive. The small deviation from the linear variation within the bend region during the accelerating ($\tilde {t}=0.25$) and decelerating ($\tilde {t}=0.75$) phases arises from viscous effects and flow curvature. The parametric variation of pressure distribution in a curved pipe at various $St$ is discussed in Appendix D (see figure 14). The cross-sectional averaged pressure $\bar {p}$ at a bend inlet is plotted over one cycle in figure 11(b) and is characterised by a discontinuity between $\tilde {t}=0$ and $1$ owing to the impulsive acceleration changing sign. The change in sign from positive to negative has an important implication in the potential for cavitation. This aspect was analysed by Wachel, Tison & Price (Reference Wachel, Tison and Price1989).

Figure 11.

(a) Variation of the dimensionless centre, inner and outer wall pressure ($\tilde {p}_s$) with distance ($\tilde {s}$) along the pipe from the inlet. The shaded region represents the bend region. (b) The variation of cross-sectional averaged pressure ($\tilde {\bar {p}}$) at a bend entry ($\phi = 0^{\circ }$) with time over one cycle. The pulsatile parameters are $Re_b = 2000$, $St=1$, and $R_c/D = 1.5$.

5.3. Forces

Figure 12(a,b) shows a decomposition of the bend force components ($\tilde {F}_x$ and $\tilde {F}_y$) into the various terms shown in (3.18) and compared with the one-dimensional analytical model (3.24). This shows the relative calculation of pressure at the inlet/outlet of the bend ($\bar {p}_{bi}$ and $\bar {p}_{bo}$), flow acceleration ($\rho \int _{V} ({\partial \boldsymbol {u}}/{\partial t} )\,\mathrm {d}V$), streamwise momentum flux changes ($\rho \int _{S_{bi}} \overline {u_\phi ^{2}} \,\mathrm {d}A$ and $\rho \int _{S_{bo}} \overline {{v^{\prime }}^{2}} \,\mathrm {d}A$), cross-wise momentum flux changes ($\rho \int _{S_{bo}} \overline {u_\phi v^{\prime }} \,\mathrm {d}A$ and $\rho \int _{S_{bi}} \overline {u_\phi v^{\prime }} \,\mathrm {d}A$) of the total bend force. These bend loading components are dominated by the unsteady pressure field for $St \gg St_{cr}$. The agreement between predictions and computations give a strong justification that the key physics is encapsulated in the simple one-dimensional model (3.24). The analysis indicates that the force components depend on the bend length ($R_c {\rm \pi}/2$) and pipe exit length ($L_d$) because these largely determine the pressure in the vicinity of the bend. The model also predicts an asymmetry between $\tilde {F}_x$ and $\tilde {F}_y$ (see figure 12c), especially near $\tilde {t}=0$ and $\tilde {t}=1$. This difference arises from $R_c {\rm \pi}/2$ and $L_d$.

Figure 12.

The dimensionless components of bend force $\tilde {F}_x$ and $\tilde {F}_y$ plotted as a function of time. In panels (a,b), $\tilde {F}_x$ and $\tilde {F}_y$ are decomposed into integral components and evaluated numerically ((3.18); solid lines) and estimated analytically ((3.24); circular markers) for $L_u/D=L_d/D=3$. (Note: (i) the components $\rho \int _{S_{bi}} \overline {u_{\phi }^{2}} \,\mathrm {d}A$ and $\rho \int _{S_{bo}} \overline {u_{\phi } v^{\prime }} \,\mathrm {d}A$ coincide to the $\tilde {F}_x=0$ axis, and the components $\rho \int _{S_{bo}} \overline {v^{\prime } v^{\prime }} \,\mathrm {d}A$ and $\rho \int _{S_{bi}} \overline {u_{\phi } v^{\prime }} \,\mathrm {d}A$ coincide to the $\tilde {F}_y=0$ axis, (ii) the plots of $\tilde {F}_x$ (sum) and $\tilde {F}_y$ (sum) with red solid lines are the summation of all the components, and the plots of $\tilde {F}_x$ and $\tilde {F}_y$ with black solid lines are the total force computed directly from FLUENT (A4) – both red and black solid lines collapse, (iii) the results are for a pulsatile inlet flow with $Re_b = 2000$, $St=1$ and $R_c/D = 1.5$). (c) A comparison between force components calculated numerically (solid, dashed and dotted lines) and analytically (circular markers) (3.26). (d) Variation of $\tilde {F}_x$ at $Re_b = 2000$ and $St=1$ with respect to $R_c/D$ is highlighted for $L_u/D=L_d/D=10$. Inset in (d) describes the discrimination between flow-acceleration-controlled ($St \gg St_{cr}$) and momentum-flux-controlled force behaviour at $Re_b=2000$ and $R_c/D=1.5$ (for $L_u/D=L_d/D=10$), and the numerical results at $St=0.01$, 0.02, 0.1, and $1$ are compared with the analytical predictions (3.27).

The influence of the curvature ($R_c/D$) on $\tilde {F}_x$ is shown in figure 12(d). The variation in $\tilde {F}_x$ arises from the enhancement or deterioration of the bend inlet pressure ($p_{bi} = \rho ({\partial {u_m}}/{\partial t}) (L_d + {\rm \pi}({R_c}/{2})$)) and bend volume flow acceleration ($\rho \int _{V} ({\partial \boldsymbol {u}}/{\partial t} )\,\mathrm {d}V = \rho ({\partial {u_m}}/{\partial t} ){\rm \pi} ({R_c}/{2}) A$). The sensitivity of $\tilde {F}_x$ to $St$ (with $Re_b=2000$ and $R_c/D=1.5$) is shown as an inset in figure 12(d). At high $St$ (flow-acceleration-controlled zone: $St \geqslant St_{cr}$ and $Re_b \gg 1$), there is a good agreement between the predictions (3.27) and numerical results because of the dominance of flow acceleration. While the slight disagreement at low $St$ (momentum-flux-controlled zone: $St \ll St_{cr}$ and $Re_b \gg 1$) is attributed to the viscous diffusion. These loading regimes (flow-acceleration- and momentum-flux-controlled zones) are highlighted in figure 1.