Drag Force on a Starting Plate Scales with the Square Root of Acceleration

We report results on the instantaneous drag force on plates that are accelerated in a direction normal to the plate surface, which show that this force scales with the square root of the acceleration. This is associated with the generation and advection of vorticity at the plate surface. A new scaling law is presented for the drag force on accelerating plates, based on the history force for unsteady flow. This scaling avoids previous inconsistencies in using added mass forces in the description of forces on accelerating plates.

Figure 1

(a) A plate with dimensions ${\ell }_a\times {\ell }_b$ moves through a viscous fluid at velocity $V(t)$ and experiences a force $F(t)$. PIV measurement plane (green) at the plate midplane. (b) Imposed velocity $V(t)$ of the plate: $V=at$ (with constant acceleration $a$) for $0<t<t_a$ and constant $V_a$ for $t>t_a$. (c),(d),(g),(h) Measured velocity fields and vorticity ${\omega }^{*}$ for $a^{*}=0.5$ ($t_a^{*}=1.0$) at $t^{*}=0.05$ (with streamlines), 1.0, 3.0, 6.5, respectively. (e) Detail near upstream plate surface. (f) Velocity profile along blue dashed line in (e); red line represents the velocity profile according to Eq. (4). The plate velocity is subtracted for (e) and (h). All dimensions are normalized by ${\ell }_b$.

Figure 2

Measured drag force $F_D$ for different accelerations toward velocity $V_a=0.45\mathrm{m}/\mathrm{s}$. Full circles indicate the end of the acceleration. The black dashed line indicates the steady state drag ($F_{\mathrm{QS}}=\frac{1}{2}{C}_D\rho A{V}_a^2$). Stars indicate the sum of the steady state drag $F_{\mathrm{QS}}$ and added mass force $m_{v}a$. Inset: peak drag force ${\widehat{F}}_D$ as a function of acceleration. Open circles: measurements not included in main figure. Full line: fitted curve ${\widehat{F}}_a\propto a^{0.5}$.

Figure 3

Measured and predicted drag force for $a=1.23\mathrm{m}/{\mathrm{s}}^2$ toward different target velocities. Full lines indicate the measured force signals, full circles indicate the end of acceleration. The star ($*$) indicates the magnitude of added mass force $m_{v}a$ [20]; dash-dotted line: $\frac{1}{2}{C}_D\rho A{V}^2+m_{v}a$; dashed then dotted line: $(\frac{1}{2}{C}_D\rho A{V}^2+F_a)$, $t^{*}\le 6$; dotted line: $(\frac{1}{2}{C}_D\rho A{V}^2+F_a)$, $t^{*}>6$. Inset: peak hydrodynamic force of the main figure as a function of $V_a$. Open circles: measurements not included in main figure. Full line: fitted curve ${\widehat{F}}_a\propto V_a^{0.5}$.

Figure 4

Instationary drag force $F_a(t)$ and dimensionless circulation ${\mathrm{\Gamma }}^{*}$ [for region indicated in Fig. 1]. Panels (a) and (c) are measurements; panels (b) and (d) follow from Eq. (5) with $C=C_a$. Colors correspond to the acceleration rates of Fig. 2; open circles indicate end of acceleration. Dashed line represents impulsively started case. Dash-dotted line at $t^{*}=6$ marks the transition to a separated turbulent wake; cf. Fig. 1.

Figure 5

Measured instationary drag coefficient $C_a$ as a function of the dimensionless acceleration $a^{*}$ for a circular disk $◯$, and square $\square$, and rectangular $⊳$ plates. Colors indicate different target velocities as in Fig. 3. Numbers in shapes represent mean values for $C_a$. Inset: drag coefficient $C_D=F_D/(\frac{1}{2}\rho A{V}_a^2)$ as a function of $a^{*}$ during acceleration for the rectangular plate. Full lines: fitted curves $F_D\propto (a^{*}{)}^{0.5}$; colors indicate $V_a$ as in Fig. 3.