We propose a design of tunable double-band perfect absorbers based on the resonance absorption in acoustic metasurfaces with nesting helical tracks and deep-subwavelength thicknesses (<λ/30 with λ being the operation wavelength). By rotating the cover cap with an open aperture on the nesting helical tracks, we can tailor the effective lengths of resonant tubular cavities in the absorber at will, while the absorption peak frequency is flexibly shifted in spectrum and the acoustic impedance is roughly matched with air. The simulated particle velocity fields at different configurations reveal that sound absorption mainly occurs at the open aperture. Our experiment measurements agree well with the theoretical analysis and simulation, demonstrating the wide-spectrum and tunable absorption performance of the designed flat acoustic device.