Low Mach, compressibility, and finite size effects of localized uniform heat sources in a gas

Abstract

The temporal evolution of the initial shock front and the low Mach regime produced behind the front due to the sudden introduction of a spherical, finite-size, low Biot number, uniformly heated energy source in a variable property gas is investigated. While the sphere is of physical interest, analogous problems of a uniformly heated infinitely long cylindrical wire and an infinite plate are also studied. Compressibility, finite-size, and nonlinear heating effects are studied without constraining the temperature of the source. Shortly after the energy source is introduced, compressibility is significant and a strong shock wave forms which weakens as it moves away from the source eventually becoming an acoustic wave. Behind it, fluid motion occurs at a much lower speed (low Mach regime), where the resulting nonlinear heating problem is solved analytically using the method of homotopy perturbation expansion leading to weak decoupling of finite-size effects and nonlinear heating effects.

Appendices

Fully compressible Navier–Stokes equations in cylindrical and Cartesian coordinates

The conservation equations for mass, momentum, and energy solved in the fully compressible Navier–Stokes [21, 34] simulations when the source of energy is an infinitely long cylindrical wire placed along the z-axis (stated in cylindrical coordinates) are, respectively,

$$\begin{aligned}&\displaystyle \frac{\partial \rho }{\partial t} + \frac{1}{r}\frac{\partial \left( r \rho u_r \right) }{\partial r} = 0 \end{aligned}$$

(45)

$$\begin{aligned}&\displaystyle \rho \left( \frac{\partial u_r}{\partial t} + u_r\frac{\partial u_r}{\partial r}\right) = - \frac{\partial p}{\partial r} + \frac{1}{r}\frac{\partial }{\partial r}\left[ 2\mu r\left( \frac{\partial u_r}{\partial r}\right) \right] \end{aligned}$$

(46)

$$\begin{aligned}&\displaystyle \rho C_v\left( \frac{\partial T_g}{\partial t} + u_r\frac{\partial T_g}{\partial r}\right) + \frac{p}{r}\frac{\partial \left( r u_r \right) }{\partial r} = \frac{1}{r}\frac{\partial }{\partial r} \left( r \kappa \frac{\partial T_g}{\partial r} \right) + \varPhi \end{aligned}$$

(47)

Here, $\varPhi = \mu \left[ 2\left( \frac{\partial u_r}{\partial r}\right) ^2 - \frac{2}{3}\left( \frac{\partial u_r}{\partial r} + \frac{u_r}{r} \right) ^2\right]$ is the viscous dissipation function. Similarly, the equations when the energy source is an infinite hot plate placed along the yz-plane (stated in Cartesian coordinates) are,

$$\begin{aligned}&\displaystyle \frac{\partial \rho }{\partial t} + \frac{\partial \left( \rho u_x \right) }{\partial x} = 0 \end{aligned}$$

(48)

$$\begin{aligned}&\displaystyle \rho \left( \frac{\partial u_x}{\partial t} + u_x\frac{\partial u_x}{\partial x}\right) = - \frac{\partial p}{\partial x} + \frac{\partial }{\partial x}\left[ \frac{4\mu }{3}\left( \frac{\partial u_x}{\partial x}\right) \right] \end{aligned}$$

(49)

$$\begin{aligned}&\displaystyle \rho C_v\left( \frac{\partial T_g}{\partial t} + u_x\frac{\partial T_g}{\partial x}\right) + p\frac{\partial u_x}{\partial x} = \frac{\partial }{\partial x} \left( \kappa \frac{\partial T_g}{\partial x} \right) + \frac{4\mu }{3}\left( \frac{\partial u_x}{\partial x}\right) ^2 \end{aligned}$$

(50)

The equation of state is $p = \rho R_g T_g$.

Fluid properties used in simulations

Simulations in this paper assume that the fluid is air and has a fixed Prandtl number (Pr) of 0.7. Air is assumed to behave like an ideal gas. For air, we have for the viscosity power law $\mu _0 = 1.716 \times 10^{-5}$ kg/m s, $T_{\infty } = 273$ K, and $n = \frac{2}{3}$. Other quantities are: Nominal Particle Radius, $R = 100$$\upmu$m, Ambient Fluid Temperature, $T_{\infty } = 273$ K, Ambient Dynamic Viscosity, $\mu _{\infty } = 1.72 \times 10^{-5}$ Pa s, Ambient Density, $\rho _{\infty } = 1.29$$\frac{\mathrm{kg}}{\mathrm{m}^3}$, Heat Capacity Ratio, $\gamma = 1.4$, Isobaric Specific Heat Capacity, $C_p = 1.005 \times 10^3$$\frac{\mathrm{J}}{\mathrm{kg K}}$, Isochoric Specific Heat Capacity, $C_v = 0.717 \times 10^3$$\frac{\mathrm{J}}{\mathrm{kg K}}$, Prandtl Number, $Pr = 0.7$, and Specific Ideal Gas Constant, $R_g = 287.058$$\frac{\mathrm{J}}{\mathrm{kg K}}$. Other quantities used that can be derived from these fluid properties are: Ambient Thermal Conductivity, $\kappa _{\infty } = \mu _{\infty } C_p/Pr = 2.46 \times 10^{-2}$$\frac{\mathrm{W}}{\mathrm{mK}}$, Ambient Kinematic Viscosity, $\nu _{\infty } = \mu _{\infty }/\rho _{\infty } = 1.32 \times 10^{-5}$$\frac{\mathrm{m}^2}{\mathrm{s}}$, Ambient Thermometric Coefficient, $\alpha _{\infty } = \kappa _{\infty }/\left( \rho _{\infty } C_v\right) = 2.972 \times 10^{-5}$$\frac{\mathrm{m}^2}{\mathrm{s}}$, Ambient Thermodynamic Pressure, $p_{\infty } = \rho _{\infty }R_g T_{\infty } = 1.01 \times 10^{5}$ Pa, and Speed of Sound in the fluid, $a_{\infty } = \sqrt{\gamma p_{\infty }/\rho _{\infty }} = \sqrt{\gamma R_g T_{\infty }} = 331.23$ m/s. The ratio of the diffusive time scale based on the thermometric coefficient and the acoustic time scales with the relevant length scale being the particle size is called the Peclet number ($Pe_{\infty }$). The nominal Peclet number is $Pe_{\infty } = 1.25 \times 10^3$.

Perturbation equations in cylindrical coordinates

On substituting (8) into (45)–(47), and (4), we obtain, respectively,

$$\begin{aligned}&\displaystyle \frac{\partial s}{\partial t} + \frac{1}{r}\frac{\partial \left( r u \right) }{\partial r} = - \frac{1}{r}\frac{\partial \left( r u s \right) }{\partial r} \end{aligned}$$

(51)

$$\begin{aligned}&\begin{aligned}&\displaystyle \frac{\partial u}{\partial t} + \frac{a_{\infty }^2}{\gamma }\frac{\partial \omega }{\partial r} - 2\nu _{\infty }\left( 1+\theta \right) ^{2/3}\frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial u}{\partial r}\right) \\&\displaystyle \qquad = -s\frac{\partial u}{\partial t} - (1+s)u\frac{\partial u}{\partial r} + \frac{4}{3}\nu _{\infty }\left( 1+\theta \right) ^{-1/3}\left( \frac{\partial \theta }{\partial r}\right) \left( \frac{\partial u}{\partial r}\right) \end{aligned} \end{aligned}$$

(52)

$$\begin{aligned}&\begin{aligned}&\displaystyle \frac{\partial \theta }{\partial t} + \left( \gamma -1\right) \frac{1}{r}\frac{\partial \left( r u\right) }{\partial r} - \alpha _{\infty }\left( 1+\theta \right) ^{2/3}\frac{1}{r}\frac{\partial }{\partial r} \left( r \frac{\partial \theta }{\partial r} \right) \\&\displaystyle \quad = - s\frac{\partial \theta }{\partial t} - (1+s)u\frac{\partial \theta }{\partial r} - \left( \gamma -1\right) \omega \frac{1}{r}\frac{\partial \left( r u\right) }{\partial r} + \frac{2}{3}\alpha _{\infty }\left( 1+\theta \right) ^{-1/3}\left( \frac{\partial \theta }{\partial r}\right) ^2\\&\displaystyle \qquad + \frac{\left( \gamma - 1\right) \mu _{\infty }}{p_{\infty }}\left( 1+\theta \right) ^{2/3}\left[ 2\left( \frac{\partial u}{\partial r}\right) ^2 - \frac{2}{3}\left( \frac{\partial u}{\partial r} + \frac{u}{r}\right) ^2\right] \end{aligned} \end{aligned}$$

(53)

$$\begin{aligned}&\displaystyle \omega - \theta - s = s\theta \end{aligned}$$

(54)

Perturbation equations in Cartesian coordinates

On substituting (8) into (48)–(50), and (4), we obtain, respectively,

$$\begin{aligned}&\displaystyle \frac{\partial s}{\partial t} + \frac{\partial u}{\partial x} = - \frac{\partial \left( su\right) }{\partial x} \end{aligned}$$

(55)

$$\begin{aligned}&\displaystyle \frac{\partial u}{\partial t} + \frac{a_{\infty }^2}{\gamma }\frac{\partial \omega }{\partial x} - \frac{4}{3}\nu _{\infty }\left( 1+\theta \right) ^{2/3}\frac{\partial ^2 u}{\partial x^2} \nonumber \\&\displaystyle \quad = -s\frac{\partial u}{\partial t} - (1+s)u\frac{\partial u}{\partial x} + \frac{8}{9}\nu _{\infty }\left( 1+\theta \right) ^{-1/3}\left( \frac{\partial \theta }{\partial x}\right) \left( \frac{\partial u}{\partial x}\right) \end{aligned}$$

(56)

$$\begin{aligned}&\displaystyle \begin{aligned}&\displaystyle \frac{\partial \theta }{\partial t} + \left( \gamma -1\right) \frac{\partial u}{\partial x} - \alpha _{\infty }\left( 1+\theta \right) ^{2/3}\frac{\partial ^2 \theta }{\partial x^2} \\&\displaystyle \quad =- s\frac{\partial \theta }{\partial t} - (1+s)u\frac{\partial \theta }{\partial x} - \left( \gamma -1\right) \omega \frac{\partial u}{\partial x} + \frac{2}{3}\alpha _{\infty }\left( 1+\theta \right) ^{-1/3}\left( \frac{\partial \theta }{\partial x}\right) ^2\\&\displaystyle \qquad + \frac{4\left( \gamma - 1\right) \mu _{\infty }}{3p_{\infty }}\left( 1+\theta \right) ^{2/3}\left( \frac{\partial u}{\partial x}\right) ^2 \end{aligned} \end{aligned}$$

(57)

$$\begin{aligned}&\displaystyle \omega - \theta - s = s\theta \end{aligned}$$

(58)

Low Mach equations in cylindrical and Cartesian coordinates

The governing equations (45)–(47) of the fluid in the low Mach regime for an infinitely long cylindrical wire become

$$\begin{aligned}&\displaystyle \frac{\partial \rho }{\partial t} + \frac{1}{r}\frac{\partial \left( r \rho u_r \right) }{\partial r} = 0 \end{aligned}$$

(59)

$$\begin{aligned}&\displaystyle \rho \frac{\partial u_r}{\partial t} = - \frac{\partial p}{\partial r} + \frac{1}{r}\frac{\partial }{\partial r}\left[ 2\mu r\left( \frac{\partial u_r}{\partial r}\right) \right] \end{aligned}$$

(60)

$$\begin{aligned}&\displaystyle \rho C_v\frac{\partial T_g}{\partial t} = \frac{1}{r}\frac{\partial }{\partial r} \left( r \kappa \frac{\partial T_g}{\partial r} \right) \end{aligned}$$

(61)

The governing equations (48)–(50) of the fluid in the low Mach regime for an infinite plate become

$$\begin{aligned}&\displaystyle \frac{\partial \rho }{\partial t} + \frac{\partial \left( \rho u_x \right) }{\partial x} = 0 \end{aligned}$$

(62)

$$\begin{aligned}&\displaystyle \rho \frac{\partial u_x}{\partial t} = - \frac{\partial p}{\partial x} + \frac{\partial }{\partial x}\left[ \frac{4\mu }{3}\left( \frac{\partial u_x}{\partial x}\right) \right] \end{aligned}$$

(63)

$$\begin{aligned}&\displaystyle \rho C_v \frac{\partial T_g}{\partial t} = \frac{\partial }{\partial x} \left( \kappa \frac{\partial T_g}{\partial x} \right) \end{aligned}$$

(64)

The equation of state is $p_0 = \rho R_g T_g$, where $R_g$ is the specific gas constant of air (287.058J / kg / K).