A method of operating a stabilizing heat exchanger of a gas turbine engine providing thermal communication between cooling air as a heat source and fuel as a heat sink to resist supercritical fluid oscillations, comprising providing a transcritical stability profile of the stabilizing heat exchanger based on a comparison between a dimensionless relationship of mean fuel pressure and dynamic fuel pressure (π f) and a dimensionless relationship of thermal energy transport and kinetic energy transport (π e) of the fuel, the transcritical stability profile defining a stable region and an unstable region of heat sink operations, determining a desired flowrate of the cooling air to be directed through the stabilizing heat exchanger that maintains the heat sink operations within the stable region according to current heat sink conditions, and bypassing a portion of the cooling air from entering the stabilizing heat exchanger to maintain the desired flowrate of the cooling air through the stabilizing heat exchanger; wherein (i) the stabilizing heat exchanger is a fuel-to-air heat exchanger having a fuel passage and a cooling air passage in thermal communication with each other, and the transcritical stability profile includes a comparison between the dimensionless relationship of thermal thermal energy transport and kinetic energy transport (π e) and the dimensionless relationship of mean fuel pressure and dynamic fuel pressure (π f) within the fuel passage of the stabilizing heat exchanger, wherein (i) the dimensionless relationship of mean fuel pressure and dynamic fuel pressure (π f) is based on a diameter of the fuel passage, a density of the fuel ( ρ), a mean static fuel pressure ( p mean), and a mass flow rate ( ṁ), and the dimensionless relationship of mean fuel pressure and dynamic fuel pressure (π f) is defined by the equation: D 4 ∗ ρ ∗ p mean m ̇ 2, (ii) the dimensionless relationship of thermal energy transport and kinetic energy transport (π e) is based on a diameter of the fuel passage, a density of the fuel ( ρ), a heat input rate ( Q̇), and a mass flow rate ( ṁ), and the dimensionless relationship of thermal energy transport and kinetic energy transport (π e) is defined by the equation: D 4 ∗ ρ 2 ∗ Q ̇ m ̇ 3; (ii) wherein the stable region includes an area below a curve on a graph having the dimensionless relationship of mean fuel pressure and dynamic fuel pressure (π f) on the horizontal axis and the dimensionless relationship of thermal energy transport and kinetic energy transport (π e) on the vertical axis, the curve being defined by the equation: π e = 8.643 ∗ π f 1.155 − 2.945 × 10 6. (i) the stabilizing heat exchanger is a fuel-to-air heat exchanger having a fuel passage and a cooling air passage in thermal communication with each other, and the transcritical stability profile includes a comparison between the dimensionless relationship of thermal thermal energy transport and kinetic energy transport (π e) and the dimensionless relationship of mean fuel pressure and dynamic fuel pressure (π f) within the fuel passage of the stabilizing heat exchanger, wherein (i) the dimensionless relationship of mean fuel pressure and dynamic fuel pressure (π f) is based on a diameter of the fuel passage, a density of the fuel ( ρ), a mean static fuel pressure ( p mean), and a mass flow rate ( ṁ), and the dimensionless relationship of mean fuel pressure and dynamic fuel pressure (π f) is defined by the equation: D 4 ∗ ρ ∗ p mean m ̇ 2, (ii) the dimensionless relationship of thermal energy transport and kinetic energy transport (π e) is based on a diameter of the fuel passage, a density of the fuel ( ρ), a heat input rate ( Q̇), and a mass flow rate ( ṁ), and the dimensionless relationship of thermal energy transport and kinetic energy transport (π e) is defined by the equation: D 4 ∗ ρ 2 ∗ Q ̇ m ̇ 3; (i) the dimensionless relationship of mean fuel pressure and dynamic fuel pressure (π f) is based on a diameter of the fuel passage, a density of the fuel ( ρ), a mean static fuel pressure ( p mean), and a mass flow rate ( ṁ), and the dimensionless relationship of mean fuel pressure and dynamic fuel pressure (π f) is defined by the equation: D 4 ∗ ρ ∗ p mean m ̇ 2, (ii) the dimensionless relationship of thermal energy transport and kinetic energy transport (π e) is based on a diameter of the fuel passage, a density of the fuel ( ρ), a heat input rate ( Q̇), and a mass flow rate ( ṁ), and the dimensionless relationship of thermal energy transport and kinetic energy transport (π e) is defined by the equation: D 4 ∗ ρ 2 ∗ Q ̇ m ̇ 3; (ii) wherein the stable region includes an area below a curve on a graph having the dimensionless relationship of mean fuel pressure and dynamic fuel pressure (π f) on the horizontal axis and the dimensionless relationship of thermal energy transport and kinetic energy transport (π e) on the vertical axis, the curve being defined by the equation: π e = 8.643 ∗ π f 1.155 − 2.945 × 10 6.